D&B toured with Blind Faith. Clapton started hanging out with Delaney, then started playing with them on stage. When Blind Faith broke up he joined D&B as one of their 'Friends", along with George Harrison. There is a 4-CD set of the tour they all did together.
Delaney convinced Clapton that he could sing and this led to Clapton's first solo album. (The one that has "Let it Rain" and "Blues Power" etc.) All of D&B's "Friends" were the band that played on that album-some of whom became Derek and the Dominos later.
One last fact for you: that first solo album by Clapton, when released, was mixed by Tom Dowd, of Allman Bros. fame. Delaney Bramlett was supposed to mix and produce the album but before he could finish it Clapton needed to return to England. The album was finished there and Tom Dowd had accompanied Clapton and some of the musicians to London. He then mixed the tapes, and much differently than Bramlett's work in progress. You can now buy a deluxe edition of this album with both mixes. They are two completely different animals.