In fruits basket why does yuki like machi?

Because besides Tohru Honda, Arisa Uotani and Saki Hanajima, she's the only girl who doesn't treat him like he's a prince and she also understands his pain of being alone and being treated like an outcast by family.

Answer:

It is because she's the ONLY one (besides possibly Saki, I mean she can see "waves") who can see past his "princely side" and see the real Yuki. Machi saw the loneliness in his smile. Tohru always treated him with a respect that made him feel lonely. That's why he tells Kakeru "It's so d*mn lonely!" And Uo-chan calls him "Prince", even at the end of the series.