Things are done falsely when they're fake or dishonest. If your sister smiles falsely at you, you'll know she's up to no good.
When people do things falsely, they are lying or otherwise being dishonorable or less than honest. You can laugh falsely to make your uncle feel better about his terrible joke, or you can fill out a job application falsely, to avoid telling a new employer how badly your last job ended. The adverb falsely comes from false, with its Latin root falsus, "deceptive, deceitful, or pretend."