It can take a long time to develop an intuition for infinite sets. Some may say that you cannot; you just calculate. This challenge applies to other areas of mathematics as well e.g. geometry in more than three dimensions. Can you visualize 4d Platonic solids? It is hard yet people have managed to classify them.
The first step is to remember what we mean by cardinality for infinite sets. Same size means that there is a bijection between them - that's it. You cannot count them in a usual sense.
An analogy that I sometimes use is a playground with a very large number of children. You want to know whether the number of boys and girls are the same. You cannot count them as they won't stay still. However, if you could get them to pair up and there was no unpaired boys or girls then you would know that there were the same number of each.
(Of course, this plan is also not very feasible in real life for many reasons. It is just a thought experiment.)
So, a bijection means that the sets are the same size without necessarily telling you want the size is.
A weirdness of infinite sets is that you can have a bijection to a strict subset. This is characteristic of infinite sets and is a possible alternative definition of infinite sets.
Back to my playground analogy. If all of the girls were paired with boys but there were some unpaired boys then you would be able to say that there were more boys (assuming only a finite number of children). In the infinite case, finding an injection that is not a bijection proves only that one set is $\le$ the other and not $<$. You need to prove that there cannot be a bijection.
Just keep on thinking of the simple map from the natural numbers to itself: $n \rightarrow 2n$. This must be the simplest example of a bijection from a set to a subset of itself. Or maybe $n \rightarrow n + 1$ is even simpler.
The next hardest bit is understanding the proof that the reals are bigger than the integers. The common proof seems to show that just one was missed. So, it is tempting to tack that on but applying the argument again shows that another was missed. We usually say that the reals are very much bigger not just a little bigger but that feeling depends on other results and is really just a feeling.
Hilbert's Hotel is a good though experiment for developing a feeling for infinite sets. Another recent question discussed it.