Solely is another word for "only" or "entirely": Your new puppy is solely your responsibility — you have to feed it, walk it, and take care of it all by yourself.
Solely comes from the Latin sola, the feminine form of "alone," and if you know other Romance languages, you'll recognize it in, for instance, Italian solo, or Spanish solamente and French seulement. In English, a solo is a song you sing alone.