Bijective map from a set to a subset of reals?

The same applies to any infinite set. In fact "infinite set" can can be defined as a set that contains a proper subset with the same cardinality. More precisely these are known as Dedekind-infinite sets.

For example take naturals $\mathbb{N}=\{0,1,2,3,\ldots\}$ and its proper subset $\mathbb{N}_+=\{1,2,3,\ldots\}$ and note that there's a simple bijection between them $x\mapsto x+1$. You can even remove infinitely many elements from $\mathbb{N}$ and still end up with the same cardinality, e.g. for $2\mathbb{N}=\{0,2,4,6,\ldots\}$ we have a bijection $x\mapsto 2x$ even though there are infinitely many elements in $\mathbb{N}\backslash 2\mathbb{N}$.

And so "being a subset" and "being equinumerous" are loosely related concepts. At most we know that $|A|\leq |B|$ when $A\subseteq B$. But $A\subsetneq B$ doesn't imply $|A|<|B|$, unless $B$ is finite.

I'm hoping someone can show me why this isn't such a strange concept?

The idea may be strange to you. Infinities are weird. But most people simply accept that and move on. There's not really anything more to do about it. After some time you get used to it and it becomes a simple fact of mathematical reality.