If each vector in a set $S_1$ is in the span of a set $S_2$, and each vector in $S_2$ is in the span of $S_1$, then they have the same span? Can someone expand on this? My book says just this one line but I'd like a bit of a longer explanation so I can understand why this is the case.
If $S_1\subseteq span(S_2)$ then it follows that $span(S_1)\subseteq span(S_2)$

To see why this is, recognize that any element in $span(S_1)$ can be written as a linear combination of elements in $S_1$, each of which since is an element of $span(S_2)$ can be written as a linear combination of things in $S_2$.

That is to say, any element in $span(S_1)$ can be written as a linear combination of linear combinations of things in $S_2$.  Thus, any element in $span(S_1)$ can be written as a linear combination of things in $S_2$ and is therefore an element of $span(S_2)$.

This proves that $span(S_1)\subseteq span(S_2)$.  The same argument can be applied in the other direction showing $span(S_2)\subseteq span(S_1)$ thus showing that the spans are equal.
Each vector in S1 is in the span of S2 so this is the span of S1 union S2. Similarly, the span of S1 is the span of hte union.