When someone acts despicably, they act in a way that deserves that hard judgment: they treat people as if they do not matter. It is the how-word for despicable, so you mostly meet it in writing and in careful speech. In casual talk, people are more likely to say “He treated her so badly” than “He treated her despicably.” It does not name the person or the act; it names the manner. You can behave despicably, lie despicably, or treat someone despicably. The feeling in the word is still moral and cold — not playful, not “a bit rude.” Though actually, because it is so strong, some speakers save it for real cruelty, and others fire it at any mean trick. Listen for the heat in the sentence.