Translation:My girlfriend is lying on the couch and listening to music.
Well, while I agree with you about "and" being unnecessary here, the grammatical structure of the sentence will change dramatically if you omit that "and". With "and", this sentence merely lists things she is doing. Without it, "listening" effectively becomes an adjective describing her state while she lying on the sofa. That would actually correspond to a different (and perhaps more advanced) construction in Russian: "Моя девушка лежит на диване, слушая музыку."
To make it more clear, compare two English sentences:
Tomorrow, I am finishing my assignment and flying to Paris.
Tomorrow, I am finishing my assignment, flying to Paris.