I've also read contradictory descriptions online, some saying that
measuring one set of entangled particles can via "spooky action at a
distance" alter the measurements of a far away set of entangled
particles. But I've also seen people basically just describe it as a
correlation due to past interaction, and that you can't actually
transfer information this way. So which is it? Can you transfer information using quantum entanglement? Or is it just a "spooky correlation at a distance"
You absolutely cannot transfer information using entanglement without the aid of an external (classical or otherwise) communication channel. This is actually quite easy to prove: given any bipartite state $\rho_{AB}$ shared by "Alice" and "Bob", and any (CPTP) channel $\Phi$ performed on Bob side (note that any possible operation can be described in such a way), if no further information is provided, the information available to Alice is $$\operatorname{Tr}_B((I\otimes \Phi)\rho)=\operatorname{Tr}_B(\rho_{AB}).$$
In other words, whatever happens on Bob's side, Alice's piece of the state is unchanged.
Still, that doesn't mean that the correlations themselves aren't nonclassical. Indeed, you can prove that quantum states can result in correlations that cannot be reproduced by any sort of classical theory.
This doesn't contradict the previous statement about no FTL communication. The correlations are indeed stronger than those allowed by classical physics/probability theory, but they also happen to unaccessible without comparing the measurement results. In other words, Alice and Bob might share a "stronger than classical link", and they can prove that they do, but they can only do so after comparing their measurement results. See e.g. Bell's theorem for dummies, how does it work? for further details. Does Bell's theorem imply a causal connection between the measurement outcomes? is also related.
I should also remark that what I'm addressing here is the nonlocality allowed by quantum mechanics. In the question, you mention entanglement. The two things are related but not quite the same. All entangled states can be used to observe Bell nonlocality, but there are entangled states which are not Bell nonlocal. Standard reference here is Wiseman et al. 2006, but going through this would probably be a bit much here. The two things are identical for pure states anyway.
Other related posts are:
This observation about the distinction between asking about entanglement and asking about nonlocality leads me to a last point, to address the titular question:
What is it about quantum entanglement that cannot be explained classically?
The phenomenon of entanglement is what you get thinking about the more general phenomenon of quantum interference/superposition from the point of view of locality. When you start asking questions about what can be achieved thanks to superpositions that could not be achieved by means of local operations only, you end up studying entanglement.
But asking what about entanglement cannot be explained classically, to me, sounds like a question that goes beyond the specific point of view that is locality. Explaining entanglement is really the same as explaining quantum interference in general. Which is essentially the same as asking what about quantum mechanics cannot be explained classically. The answer to which is: any phenomenon that is not compatible with classical explanations. There's plenty, so this becomes a bit broad for a single post.
