Something that's done angrily happens in an annoyed or hostile way. If you slam the door angrily, you do it because you're furious.
When you speak angrily to someone, your displeasure is obvious in your voice, and if your neighbors react angrily to your decision to start keeping bees in your backyard, they are deeply unhappy about it. The adverb angrily comes from its related adjective, angry. The Old Norse root is angra, "to grieve, vex, distress, or take offense with," from Proto-Germanic origins.