Thusly is a superfluous word. Because thus is an adverb in its own right, the adverbial -ly adds nothing. This doesn’t mean that thusly is wrong, however, and there are contexts in which many English-speakers find it simply sounds better than thus, especially where it introduces quotes or lists.
Thus could replace thusly in each of these sentences:
Cliff thusly became the first pitcher since Deacon Phillippe to walk no one and strike out ten men in a complete game World Series victory. [Patch]
Being thusly aware, I am also aware of the various levels of irony attendant in what I am about to say. [ABC Online]
Thusly, the beautiful boho bags from East Village label, Elliot Mann, have the luck of feeling utterly of-the-moment. [NBC New York]
And just to prove that thus works as an adverb, here are a few examples:
The newspaper is thus compromising its own integrity. [The Nation]
Mr Cameron was thus obliged to spend several unscheduled hours in the company of the senior British officers. [Telegraph]
They also invite pictures, video and other contributions, … thus incorporating non-journalists into the news system. [Economist]
Thusly is most often used to introduce quotes or lists—for example:
Instead, Orlando Winters of the Miami New Times defines it thusly. [Jezebel]
Hence, the question might be posed thusly: “Shall the law take effect?” [Columbus Dispatch]
While thus technically could replace thusly in these instances (and we could present many examples of thus used this way), this use of thusly is common and probably doesn’t bother many people.