A topological group is a group $G$ whose underlying set is equipped with a topology such that:
$(1)$ the multiplication map $\mu: G \times G \rightarrow G,$ given by $(x,y) \rightarrow xy$ is continuous if $G \times G$ has the prouct topology.
$(2)$ the inversion map $I : G \rightarrow G$, given by $x \rightarrow x^{-1}$ is continuous.
I don't understand how a function between the two groups $G \times G$ to $G$ can be continuous. How do you define a continuous map between two groups?
If you look at your statement, it says "is equipped with a topology . . ."
So, the way you think of a continuous map $G\times G \longrightarrow G$ is the same as you think of any continuous map, that is, the preimage of any open set is open. You may want to consider that in this case, you are looking at the product topology on $G\times G,$ and then you can use what you know about the definition of the product topology to help you out.