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.