Politically means “in a political way,” or “when you look at it as a matter of government, parties, and public power.” “Politically, the plan is difficult” means the problem is not the science or the money — it is votes, parties, and public feeling.

Though actually, the word does a lot of work in set phrases. Politically correct (often shortened to PC) means careful not to offend groups of people; that phrase has become a fight of its own, and older and younger speakers do not always say it with the same feeling. Politically active means you take part in politics. Politically motivated often hints that the real reason is power, not the reason people say out loud. Politically charged means the topic makes people angry along party lines.

It is the everyday -ly form of political. The cousin politic (wise / careful) almost never makes an adverb in real life; people say “tactfully” or “wisely” instead.