Translation:I have not eaten meat since I became vegetarian.
That would be "siden jeg er blevet vegetar".
By the way, wouldn't the English sentence sound more natural with "a vegetarian"?
here the word vegetarian is being used as an adjective, so no 'a' is needed. You could change it to be a noun though, and it would mean the same thing: 'a vegetarian'.
You're right. It isn't an adjective in the Danish sentence though, since that would be "vegetarisk".
Regardless of what's happening here in Danish, in English the "a" is necessary. It simply sounds weird to say "since I became vegetarian".