When the big bang actually occured, well we can use the following as a great example of how it expanded.

For instance, take an unbaked chocolate chip cookie (yum...) for example. Prior to it being baked, you have a small area with chocolate chips in it (all within a small area). However, when you shove that delicious cookie in the oven the entire area expands, as well as the chocolate chips move apart from one another. Now as a final product you have a multiple of more room since it was baked (more area), and your chocolate chips are spread out moreso than from the start.
**This is my artistic, ha, illustration of what I'm talking about**

In a way this is how our universe works too. Instead of the oven stopping though, so far it just keeps on expanding and hasn't stopped yet. So every day our universe is gradually getting bigger and giving more room for other galaxies to move further and further apart (and sometimes maybe even collide) but they are given more freedom room to move.