Monsieur Dambreuse is all about the Benjamins. And the power that comes with 'em. Without fail, he'll get behind any regime that supports his wealth and business—no matter the politics.
He and his PYT are well-connected high rollers who are always hosting some shmancy party. Early on in the novel, Deslauriers actually recommends that Frederick hook up with them. You know, for networking:
"You ought to ask that old chap [Mr. Roque, the neighbor] to introduce you to the Dambreuses. There's nothing so useful as to be a visitor at a rich man's house. Since you have a black coat and white gloves, make use of them. You must mix in that set. You can introduce me into it later. Just think!—a man worth millions! Do all you can to make him like you, and his wife, too. Become her lover!" (1.2.44)
Frederick follows this advice, in his own slow way, and eventually takes a major plunge when he gets engaged to Madame Dambreuse approximately six minutes after her husband dies. To be fair, though, Frederick had always had his eye on her. Let's rewind to chapter 1 for a peek:
All he could see was her back, covered with a violet mantle. However, he took a glance into the interior of the carriage, lined with blue rep, with silk lace and fringes. The lady's ample robes filled up the space within. He stole away from this little padded box with its perfume of iris, and, so to speak, its vague odour of feminine elegance. (1.3.15)
Lust, curiosity, and a desire to improve his social status and come into some serious cash. Always the basis of a strong relationship. And of course, Frederick changes his mind when he realizes that (a) she really has no money (her husband left it all to his illegitimate daughter, Cécile), (b) she's shallow, and (c) she's cruel, buying one of Madame Arnoux's possessions just to torture him.
In a not-at-all twist ending, Frederick leaves her high and dry.