Remember how "agreeable" he made himself to all of the Dashwood ladies? None of them wanted to think ill of him, even when he ran off. Mrs. D was appalled that Elinor would even consider questioning his obvious amiability and high character.

So at some point he has met Miss Grey, and made himself highly agreeable to her as well. If she was not any great stunning beauty, she was probably flattered - he seems smooth enough to make it appear that he liked her for herself, not just her fortune.

And then she sees Marianne at the party - well, Willoughby probably had enough smooth talker chops to cover that over fairly convincingly, especially if Miss Grey did not want to go through the stress of breaking the engagement. From the way he wrote the evil letter, it sounds like he may have even half-convinced himself that he was just "nice to the family" at Barton Cottage and that Marianne read more into it than was there. If he could tell himself that lie well enough to write such a letter, he could surely tell it to his betrothed well enough to convince her that Marianne was a silly little country girl with an unrequited crush that he had not meant to encourage. She still gets her landed gentleman, he still gets the $$$, and everyone gets to pretend that nothing inappropriate has happened.

On the other hand, if she breaks the engagement she has to admit to herself and all the world that W does not love her so well as she believed and that he deliberately flirted with a much prettier girl. Even though she could break the engagement without breach of contract or other serious material consequences, there would still be social consequences of being known to be a sort of "rejected" woman, I imagine. So (in my imagination, at least) she believed what she wanted to believe and what was convenient to her.