When I first came across this problem (albeit worded differently) my trouble in solving it was simple: I didn't know that, by definition, all words are of finite length. There are no words of infinite length. Perhaps this helps you? Read on otherwise...

As all words are of finite length this means we can list the words by their length.

There are countably many words of length for ( to be precise). Do this for all lengths, and so the number of elements in your group is a countable sum of countable numbers. This is a countable number times a countable number. This is countable.

I hope that makes sense. I also apologise for my sloppy use of words such as "countable" and "number". I'm never entirely sure when to use what...

As an important aside: this means that any countably (infinite or finite) generated group is countable.