Validly describes something that's done in a reasonable or legitimate way. If you spy your little sister with chocolate on her face, you can validly assume she's the one who ate the last brownie.
There are two basic ways to use this adverb — first, you can substitute it for a slightly more formal version of "reasonably," like when you validly conclude that the author of the book you're reading never actually travelled to India. You can also use it for something that's done in a legally binding way: "The will wasn't validly signed, so your grandmother's dogs won't inherit her house after all."