There are a number of reasons. For one thing, their respective characters and personalities really couldn't be more different. Blanche sees herself as a traditional Southern belle—elegant, refined, and educated. Stanley, on the other hand, is a hulking great brute of a man: a regular blue-collar guy who doesn't take kindly to being condescended to by Blanche.
The socio-cultural gulf between the two is an additional factor in their burning animosity. Stanley is a Polish-American, and this makes him an object of prejudice for Blanche Dubois, whose French surname indicates a much more noble heritage. But irrespective of his background, and whatever his many faults, Stanley also has a certain rugged honesty about him, and this makes him intensely suspicious of the secretive Blanche.
Then there's the little matter of Stella. To some extent, Blanche and Stanley are engaged in a battle of wills over Stella's soul. Blanche wants to rescue her sister from an increasingly abusive relationship with a man she regards as little better than an ape. But Stella's staying put—not out of fear, but because she genuinely loves Stanley, and there's absolutely nothing Blanche can do about that. The situation's made worse from Blanche's point of view when Stella gives birth to Stanley's baby. This brings Stella even closer to Stanley, and by the same token, pushes her further away from Blanche.