This is easiest to understand from the point of view of someone standing along the tracks. The train is accelerating forward. The man is accelerating in the direction of the train.
Why is the man accelerating? He is touching the floor. Friction is the force the floor exerts on the man to keep him from sliding. The floor pulls him forward.
Suppose another man was sitting on a block of frictionless ice. When the train accelerated, the man would not move. The train would move out from under him, and he would fall off the back end.
One characteristic of forces like static friction is they are just big enough to do the job. Another example of such a force is the force the floor exerts on the man to keep him from falling through the floor. Gravity tries to accelerate a man downward. The floor exerts an upward force just strong enough to prevent this. If the man carries a weight, the gravitational force on the man plus weight increases. So the floor pushes upward harder to keep the man plus weight from penetrating the floor.
Another characteristic of these forces is they can have limits. A really heavy weight would break through the floor.
For static friction, the floor usually keeps the man from sliding as the train accelerates. This means the man accelerates at the same rate as the train.
There is a limit to how strong static friction between the man and the floor can be. If the train accelerates strongly, static friction will not be strong enough. The man will accelerate in the direction of the train, but the train will accelerate at a higher rate. The man will slide toward the back of the train and fall off the end. The problem is asking you to calculate how big the acceleration of the train must be for this to happen.
The confusing thing is related to the fact that that the train is accelerating forward, but the man falls off the back end.
People often do not look at the world from the point of view of the man standing by the tracks. The conductor thinks of the train as motionless, and the world outside as moving toward the back of the train. This is reasonable. The conductor stands by a seat. A while later, the seat is still right next to him. It has not moved.
A conductor can do physics in an accelerating train, but a complication is added. It appears to the conductor that a force pulls everything in the universe toward the back of the train. The conductor sees the man sitting on a block of ice accelerate toward the back and fall off the end.
This force is given the unfortunate name of "fictitious force", as though it doesn't really exist or doesn't have a cause. It does exist. The conductor really does see the man on ice accelerate toward the back. It exists because the conductor is ignoring the forward acceleration of the train so he can treat the train as motionless. Doing so makes him treat the man on ice as occupying positions farther and farther toward the rear of the train.
The conductor's point of view sounds wrong when described like this, but it is a perfectly reasonable thing for him to do. It easy for him to keep track of a motionless seat in a motionless train. It would be a lot harder to calculate his accelerating position, the seat's accelerating position, and determine that the distance between him and the seat has not changed. If he must add a fictitious force to everything, that is a small price to pay.
Now we can repeat the problem from the conductor's point of view.
The train accelerates forward slowly. The fictitious force pulling the man on ice toward the back gives him an acceleration is just as big as the acceleration of the train that the conductor is ignoring. The man standing on the floor is also pulled backward by this fictitious force, but friction from the floor pulls him forward just as strongly. He stays motionless.
The train accelerates more strongly. Friction is not strong enough to hold the man standing on the floor motionless. He slides toward the rear of the train.
