In the spirit of this answer to a different question, I'll offer a contrarian answer. How to understand probability theory from a structuralist perspective:
Don't.
To put it less provocatively, what I really mean is that probabilists don't think about probability theory that way, which is why they don't write their introductory books that way. The reason probabilists don't think that way is that probability theory is not about probability spaces. Probability theory is about families of random variables. Probability spaces are the mathematical formalism used to talk about random variables, but most probabilists keep the probability spaces in the background as much as possible. Doing probability theory while dwelling on probability spaces is a little like doing number theory while dwelling on a definition of 1 as $\{\{\}\}$ etc. (That last sentence is definitely an overstatement, but I can't think of a more apt analogy offhand.)
That said, multiple perspectives are always good to have, so I'm very happy you asked this question and that you've gotten some very nice noncontrarian answers that I hope to digest better myself.
Added: Here is something which is perhaps more similar to dwelling on probability spaces. To set the stage for graph theory carefully one may start by defining a graph as a pair $(V,E)$ in which $V$ is a (finite, nonempty) set and $E$ is a set of cardinality 2 subsets of $V$. You need to start tweaking this in various ways to allow loops, directed graphs, multigraphs, infinite graphs, etc. But worrying about the details of how you do this is a distraction from actually doing graph theory.