I think that part of the difficulty (and part of the reason that certain people call themselves infinite set atheists) stems from the fact that we have two very basic intuitions about the quantity of finite sets, and it is impossible to define quantity for infinite sets in a way that maintains both intuitions.

Namely, you can have a notion of quantity for which

(A) sets that can be set in some 1-to-1 correspondence will have the same quantity,

OR a notion of quantity for which

(B) a set that strictly contains another set will have a strictly larger quantity.

As it turns out, given the importance of functions and correspondences in basic mathematical questions, the formulation (cardinality) that preserves (A) is very natural for doing math that extends and coheres with other finite intuitions, while only a few logicians seem to toy around with (B).

So it may help to realize that for mainstream mathematics and its applications, there is no way to rescue (B); you'll just need to get used to the idea that an infinite set and a proper subset can have the same cardinality, and the notion that what matters is the equivalence relation of there existing some 1-to-1 correspondence between sets.