'Grrrl' is an alternate word for 'girl'. It refers to a woman who is admired for being independent, or creative, or strong. It is pronounced the same as 'girl', but with emphasis on the -ir- part of the word. This emphasis is done by lengthening the -ir-, so that the word takes about 3-4 times as long to say. The reason why it's written with multiple r's is that most of the lengthened -ir- sound is with the 'r'.
A couple of other things: There is no difference between 'grrl' and 'grrrl', just alternate spelling. Also, there is another alternate spelling, and that is 'gurl' or 'gurrl'. I suspect that's what you heard.