Like Branr said, you will need a food bowl to put the gourmet food in for you pet to eat. It doesn't matter what kind of food bowl, and if you don't have a lot of money even the Cheapo food bowl will work. Other wise, instead of clicking Fill (10$) Click the bowl for the Fill With Gourmet Food option or click on the gourmet food itself. I'm pretty sure both ways work, if I can remember. Another easy thing to do is save up your pet's Life Time Happiness Points and get a bottomless pet bowl, so not only do you have to worry about refilling, it also gives your pet a 10+ moodlet every time it eats.
~Flower