Something that's meaty is dense and chewy, or full of rich flavor, like meat. Believe it or not, a meaty portobello mushroom can be a good substitute for a hamburger.
If you serve your guests a thick, meaty stew, that means it's actually full of chunks of meat, and when you cook a pan of vegetarian "bacon," it may smell so good and meaty that you tempt even your most carnivorous friends. A figurative way for something to be meaty is to be full of information or substance. A meaty essay will make your history teacher much happier than one that's thin and hastily written.