When Elizabeth Bennett's sister Jane is urged to visit Netherfield Hall by their mother, it rains and she falls ill. Naturally, Mr. Bingley convinces Jane to stay at their house until she is well.
Elizabeth and Jane, being the closest in age to each other, are very close. Elizabeth is worried about Jane and goes to Netherfield Hall to chack on her. Elizabeth walks the entire way, and arrives in a somewhat bedraggled state. But when Bingley's sister and the other ladies gossip about how awful and muddy she was, it's clear that Mr. Darcy was somewhat taken by her appearance.
As Elizabeth's visit goes on, Mr. Darcy starts to see what an intelligent person she is. She teases him, and although he doesn't give her much response, it's clear that he is looking at her in a new way. He likes that she has her own thoughts and opinions and isn't shy about sharing them.
Although the two meet and interact on other occasions, this part of the book is the pivotal time when we (the readers) can see Darcy falling in love.