Monsters have huge appetites so check your health level icon on the left side of your room screen. If your monster's health is under 999, it's probably hungry!

To feed your monster, head to the Gross-ery Store on Main Street and purchase some snacks. Check the inventory in your room to find your purchases, and then drag and drop them onto your monster's belly. 
Your monster will let you know which food it likes best. Will your monster be a fan of Roast Beast or Sunshine Berries? And how about washing that down with a Cup o' Gloop. Mmmm yummmm.

TIP: Be sure not to over-feed your monster!