Once you've completed the beast training quest, which if memory severs is basically a trip from to Darnassus from Dolanaar, you'll get some new abilities in your Beast Mastery tab. Mend Pet, Revive Pet, Feed Pet, and I think one other but the name escapes me at the moment.
To feed your pet you just click Feed Pet then click on a piece of food that your pet will like. Now it is important to feed them food that they like. You can find this out by bringing up your character page and clicking on the pet tab. In the left hand corner you'll see the happiness indicator, mousing over this will tell you your pet's diet.
I too got confused by this as well when I first got my pet.