How can I listen to a playlist on iPad?
As you've discovered, the Remote app can really only be used to remotely control playback on other devices.

If you don't wish to sync the playlists from your computer to your iPad via iTunes, then you'll have to stream them some other way. Unfortunately, for whatever (probably music industry-enforced) reason, Apple's own AirPlay technology does not allow sending audio from a Mac or Windows PC to an iOS device. Fortunately, there are 3rd-party solutions that do allow this, like AirFoil.