How did the Dursleys know Mrs. Figg?

Question

What I really want to know but don't want to put in the title because it could be considered a spoiler:

Did the Dursleys know she was affiliated with the Wizarding community? I believe we are told Dumbledore tells Mrs. Figg to watch over Harry; did he also tell the Dursleys she could take Harry from time to time? Given the description of the Dursleys, I'm surprised they would willingly expose themselves to more of their kind.

Did the Dursleys just meet her and figure their crazy nephew would get along perfectly with the crazy cat lady?

Answer 1

We have no indication that the Dursleys knew that Mrs. Figg was in any way associated with the Wizarding community, but it's most sensible to assume they did not. As you point out, it would be entirely out of character for the Dursleys to willing place Harry in the care of anyone magical, no matter how remotely, especially when they were explicitly trying to "stamp [the magic] out" of him:

"I'm not having one in the house, Petunia! Didn't we swear when we took him in we'd stamp out that dangerous nonsense?"

^{Harry Potter and the Philosopher's Stone Chapter 3: "The Letters from No One"}

Moreover, when we actually get to speak with her Mrs. Figg is quite clear that Harry being miserable was considered a bonus by the Dursleys, and is at least part of why he was allowed to spend time with her:

I'm sorry I gave you such a miserable time, but the Dursleys would never have let you come if they’d thought you enjoyed it.

^{Harry Potter and the Order of the Phoenix Chapter 2: "A Peck of Owls"}

Similarly, it seems unlikely that Dumbledore told them about her; he convinced the Dursleys to take Harry in mainly by appealing to familial guilt, but it's clear they had no qualms about ignoring his instructions when it suited them. If Dumbledore had told them "if you ever need a quiet weekend, give the boy to the weird lady a few streets down with all the cats," it seems most likely that they'd have done exactly the opposite, if for no other reason than to avoid exposing him to anything even slightly magical.

Remarkable as it may seem, there was once a time when people talked to their neighbours¹; since she lived so close to Privet Drive, it doesn't seem entirely unreasonable for the Dursleys to know her socially. How they first met, however, is unknown to my knowledge.

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¹ And I say this as someone who has met maybe two of my many neighbours; no judgement, trust me

Answer 2

Mrs. Figg was a member of the Order of the Phoenix. She was there on Dumbledore's orders with explicit instructions to watch over Harry. That being the case, in order to do her job, she presumably went out of her way to meet the Dursleys and to make sure that they liked her. She admitted to giving Harry a hard time to make sure that they would send Harry to visit, which showed that she was going out of the way to make sure that she'd be part of Harry's life.

Presumably, she did not tell them about her association with the magical world. Harry at least was surprised by the association even after having spent several years at Hogwarts, so she was pretty good at hiding it (or Harry didn't put two and two together). Given their hatred of anything "out of the ordinary" or associated with magic, I can't imagine the Dursleys would be too eager to have anything to do with her if they knew about the connection (especially given how hard they had worked to make sure that Harry never found anything out).