First of all, I have to point out that there ought to be a distinction made between civil liberties (things like First Amendment rights) and civil rights (14th Amendment/Equal Protection). I know that people often use them interchangeably, but there is a difference.
King's letter is really about civil rights. Its importance for civil rights is that, as the link below says, it
is credited with establishing the religious and intellectual foundation for civil rights.
In this letter, King lays out why he is fighting for civil rights. He talks about why it is important that all people should have civil rights and why they should not have to wait to get them. He also lays out his reasons for pursuing civil rights through nonviolent civil disobedience.
This letter has come to be seen as the classic statement of why and how people should fight for their civil rights.