It's a little tricky to talk about the opening lines of the Compromise, because in the end it was passed in different sections. So what counts as the opening lines? For the sake of clarity, though, we'll go with the first line of the whole business:

It being desirable, for the peace, concord, and harmony of the Union of these States, to settle and adjust amicably all existing questions of controversy between them arising out of the institution of slavery upon a fair, equitable and just basis. (Resolutions.Section 1.1)

This line actually nicely sums up why we're all here reading about all of this. This first line admits that the whole Compromise is about slavery, and how to deal with it in a way that's fair to everyone. Except, of course, the slaves.

It's a practical document, with a very practical purpose, so it opens with a succinct description of what its purpose is. If only it really did solve "all existing questions" about slavery. It took a little more than some debate in the Senate to do that in the end.