It has to do with various contracions of the "I am" construction. I know how to recognize them but can't give you all the ins and outs of what they are contracted from. As I understand it though, if you spelled out all the words, it would be something like - Rydw i yn dysgu Cymraeg.
However, whenever you have the "yn(+verbnoun)" following a vowel (as you do here) you always contract it to 'n - Rydw i'n dysgu Cymraeg.
I have a button from the Welsh Language Board that contracts that further to - Dw i'n dysgu Cymraeg
- and I know there are a lot of other ways to contract the form. I'm sure that Rwy'n dysgu Cymraeg
is simply another way to contract the same sentence.
Now let's see who will correct me, or at least expand on what I've said!