The exercise to make everything clear
Let me take a simpler example. Suppose we're talking purely 1D classical mechanics, and you have two particles, one moving with constant speed $u$ and the other one moving with a time-varying speed parametrized by $\alpha$, $$v(t) = \frac{\alpha~u~ t}{\sqrt{u^2 + (\alpha~t)^2}}.$$
We start off the first one at position $x=0$ and the second one at position $x=L$ and set them both moving with these speeds in the $+x$ direction.
Exercise: prove that there is some $L$ such that they will never meet, that the constantly-decreasing speed difference means that the time when they actually meet goes to $\infty$.
What? Why does that clarify anything?
Now what does this have to do with relativity? Well it turns out that the above $v(t)$ is the $v(t)$ prescribed by the relativistic rocket equation, if $u = c$. Therefore from the standpoint of the person on the ground, a light ray never catches up with the relativistic rocket if it is emitted behind a certain distance.
(What does that look like from the perspective of the rocket? It looks like there is a black-hole-like event horizon behind them; these events are being emitted from behind that horizon.)
The rest of the paradox is just a standard confusion at a sort of tension between two ideas in relativity: relativity enforces some form of speed limit where you would need an infinite amount of energy to go faster than speed $c$, but nobody actually experiences this limitation as a limitation. The very principle of relativity is that nobody is ever saying, "here is my absolute motion through absolute space" and so people must experience themselves as being at rest, with light still moving away from them in all directions at speed $c$ in their rest-frame, even though it also recedes at speed $c$ in other reference frames that are moving relative to me. But this is the standard relativistic tension.
Some tension-relief: it's all about the simultaneity.
I guess my first remark that I like to make to people starting this out is, please pay careful attention to the first-order Lorentz transform. Every other Lorentz transform in relativity can be understood as the limit of "do a little Lorentz transform, then do another little Lorentz transform, then do another little Lorentz transform... repeat until you've built up your whole transform." And this is where we see a difference with Newtonian mechanics. Transform by any $\vec v = [v_x;~v_y;~v_z]$ large-or-small, and you will find that Newton prescribes that you convert time coordinates $t$ and space coordinates $\vec r = [x;~y;~z]$ according to:$$
\begin{array}{rcl} t' &=& t,\\\vec r' &=&\vec r ~-~ \vec v ~ t.\end{array}
$$However Einstein's crazy idea is that this does not represent what's actually happening when we accelerate relative to each other. Instead for small velocities $\vec \epsilon = \vec v/c,~|\vec \epsilon| \ll 1,$ you should transform your time coordinate $w = c~t$ and your space coordinates $\vec r$ to the values: $$
\begin{array}{rcl} w' &=& w ~-~ \vec \epsilon \cdot \vec r,\\\vec r' &=&\vec r ~-~ \vec \epsilon ~ w.\end{array}
$$
Note that the second equation is essentially unchanged but the first equation now "looks like" the second equation in that the velocity is coming in to 'desynchronize' our clocks. This is called the "relativity of simultaneity" and I am just mentioning in passing that the other two core effects of relativity, "length contraction" and "time dilation", really come from adding up the way that these little desynchronizations get transformed and retransformed by these above equations. Then it gets more useful to confine the motion to one particular direction, like $\vec v/c = [\beta;~ 0;~ 0],$ in which case you can write that the general result is:
$$
\begin{array}{rcl}
w' &=& (w ~-~ \beta~x)~/~\sqrt{1 - \beta^2},\\
x' &=& (x ~-~ \beta~w)~/~\sqrt{1 - \beta^2},\\
y' &=& y,\\
z' &=& z.
\end{array}
$$
So, that's what happens when you propagate a ton of these little desynchronizations from little transforms through each other into a bigger transform. These terms $1/sqrt{1 - \beta^2}$ appear that start to affect "here is how long I think your spaceship is, it's much shorter than you think!" You can derive them a totally different way, by seeing that the cross term $2~\beta~w~x / (1 - \beta^2)$ is the same for the square of both, so that $(w')^2 - (x')^2 = w^2 - x^2.$ This is very important because this property that $\big(w^2 - x^2 - y^2 - z^2 \big)' = w^2 - x^2 - y^2 - z^2$ is really a statement about everyone agreeing on the speed of light receding from you: imagine that an event happens suddenly at your location, the light notifying the rest of the cosmos about that event now recedes from you as a thin spherical shell at the speed of light, which means obeying $\sqrt{x^2 + y^2 + z^2} = c~t.$ Well we defined earlier that $w = c~t$ and so this number $w^2 - x^2 - y^2 - z^2$ is zero for this expanding-light-bubble: and what we've just proven is that zero-for-me means zero-for-everyone; nobody sees this "light cone" that notifies the universe about this event as anything other than a perfect sphere expanding with velocity $c$. So this is what these little desynchronizations are really doing in the background, they're making everyone agree on the structure of these light bubbles, which makes everyone agree on things like causality and stuff.
Now be brutally honest about spaceships.
My second remark is that we have to think very brutally and physically about what we are describing, if we want to understand what's really going on. What happens to the spaceship that keeps accelerating faster and faster but never hits the speed $c$? Well, let me model its constant acceleration in a simple way: it drops some sort of "beacon" and then accelerates until it detects that beacon is travelling away from it with speed $u,$ then it throws down another beacon and accelerates until that beacon travels away with speed $u,$ and so on, and so on. To accelerate with constant acceleration, the clock where these beacons are thrown down must be throwing them down at constant intervals.
Combine those two last paragraphs and you realize that trying to outrace a beam of light is a real-life Zeno paradox! You accelerate to $c/4$ relative to this beacon, measure the speed that the beam of light is moving away from you, and it's still moving at speed $c$ away. You throw down another beacon, accelerate to $c/4$ relative to that, measure the speed that the beam of light is moving away from you, and it's still moving at speed $c$ away. You cannot win! Therefore nobody can travel faster than the speed of light. These bubbles of light then enforce a very powerful form of causality that's not technically present in Newtonian mechanics; it says: no form of matter can get to a destination faster than the information about that matter's history, which moves at speed $c$.
Then, learn to love rapidities.
Now when the spaceship tries to measure all of the beacons as a whole, it does not see a straightforward linear progression of speeds: those speeds would go off to infinity, therefore they would exceed $c$! Rather, it sees a progression which goes like a function called the hyperbolic tangent, $\tanh x = (e^x - e^{-x})/(e^x + e^{-x}).$ In fact if we stick with motion in one dimension we find out that each velocity can be characterized by a rapidity, $\beta = \tanh \phi,$ and that the velocity addition formula actually then becomes linear in rapidities. So that's why you see this hyperbolic tangent pattern in the speeds of the beacons! It's because your single-beacon Lorentz boost shifts its rapidity by some $\phi_1$, then every other rapidity must be $\phi_n = n ~ \phi_1,$ therefore every velocity must be $v = c \tanh (n ~\phi_1),$ for cases of constant acceleration.
Finally, one integral gets the original formula.
Actually rapidities are even more useful than that, because our expression $\gamma = 1/\sqrt{1 - \beta^2} = (e^\phi + e^{-\phi})/2 = \cosh \phi$ by the variant of the Pythagorean theorem that holds for hyperbolic trigonometry, $\cosh^2\phi - \sinh^2\phi = 1.$ This lets us perform the following integral. We know that the spaceship emits beacons at some interval of proper time $\delta \tau$ and can easily reconstruct the time on the clock $\tau = n~\delta \tau$ from that plus the number of emitted beacons. Since each time the spaceship gains a constant boost to its rapidity $\delta \phi,$ making everything exact to first-order gives the straightforward result that the speed of the spaceship is $v = c~\tanh(a~\tau/c).$ But our time coordinates $t$, watching all of this, are going to be different than these $\tau$ coordinates in the spaceship! Well that's where we just integrate: the basic time-dilation formula says that $dt = \gamma~d\tau = \cosh(a~\tau / c)~d\tau.$ Integrating that and discarding the constant gives $t = (c/a) ~ \sinh(a~\tau/c),$ therefore its velocity using our time coordinates for our reference frame is simply:$$v(t) = c\tanh\big(\sinh^{-1}(a t/c)\big).$$But, one of these basic results from $\cosh^2 \phi - \sinh^2 \phi = 1$ is that $\tanh \phi = \sinh \phi / \cosh \phi = \sinh \phi / \sqrt{1 + \sinh^2 \phi},$ so therefore $\tanh\big(\sinh^{-1} x\big) = x / \sqrt{1 + x^2},$ and the above simplifies to,$$
v(t) = \frac{a t}{\sqrt{1 + (at/c)^2}}.
$$
Multiplying both top and bottom by $c$ gives precisely the form in the above exercise I provided with $u = c.$ Then we have a perfectly valid reference frame in our ground reference frame, and we know that this spaceship moving with speed $v(t)$ will not be hit by some incident beams of light with speed $c$, if they start out a distance $L$ behind it. All of the rest of the reasoning is identical between classical mechanics and special relativity; special relativity only affects how you change between reference frames.
Quick recap
To recap, because the spaceship is emitting these beacons at constant time intervals, with no concern about its own state; and because the universe has the particular desynchronizing structure it has, we discover that we at rest don't think that the beacons are emitted at constant time intervals, nor is the spaceship linearly accelerating. Instead we measure the beacons as actually emitted slower and slower, and we measure the velocity of the spaceship as a $v(t)$ curve which as we learn from the exercise above, cannot be hit by light rays from a certain distance $L$ before where it started. Therefore we realize that when you accelerate, there is some strange event horizon created at a very long distance behind you, and it gets closer and closer as you accelerate harder and harder. Events from behind this "wall of death" contain information that cannot possibly reach you; objects that you pass appear to slow down as they approach the wall of death, getting slower and slower as they fade closer and closer into the wall, never quite falling all the way through. You can only get rid of it by ceasing to accelerate, which will push that wall of death further and further away.
Just to complete the reasoning, John Rennie is claiming that one of the twins must see this wall of death and cannot use an inertial reference frame with special relativity to describe his/her situation; instead he/she needs to use general relativity with the Rindler metric, which basically forces him/her to acknowledge a form of gravitational time dilation is applying specially to him/her. Thereby, both twins agree again.
