Why is there so much animosity between Stanley and Blanche?

The animosity that permeates the Kowalski household following Blanche's arrival is largely driven by threat that Blanche poses to Stanley's dominance, specifically in relation to his control over Stella. During the first few lines of the play, the reader is granted an everyday view of the relationship between Stella and Stanley. His first words to his wife are 'bellowed' and he continues to speak loudly and brusquely to her even when she asks him not to. When he throws the bloody package of meat to/at her, ignoring her protest, it becomes clear that Stanley is the one in charge in their relationship. He acts without considering his wife's feelings or opinions and, furthermore, expects her to submit to those actions. This dynamic is immediately threatened when Blanche arrives.
Blanche is not only Stella's older sister, but her only family and, as such, is perhaps the only person who could challenge the power that Stanley holds. As Blanche settles into the Kowalski house, Stella's focus moves from serving her husband to serving her sister. A clear instance of this is seen at the beginning of scene two, when Stanley comes home to find Stella preparing to take Blanche out for a night on the town. Stanley is immediately turned off by the fact that Stella has simply left him "a cold plate on ice" instead of a warm meal on the table, which he is probably very accustomed to. Furthermore, Stella tells Stanley to compliment her sister when she comes into the room because "it's important to Blanche". Already Stella is putting her sister's needs over her husbands. Throughout the rest of the play, Stella allows Blanche to hold more and more power in the Kowalski household. Blanche has unrestricted and unlimited access the bathroom and bedroom of the small abode and in several instances, Blanche tries to restrict Stanley's access to these same areas. Stella also confronts Stanley often throughout the play over his treatment of Blanche, which goes against the very foundation on which their relationship is built.
The tension between Stanley and Blanche finally comes to a head during scene ten, when Stanley informs Blanche that he knows exactly how she lost Belle Reve and why she left her hometown. His exposure of her promiscuous past is the first step in exerting his dominance over her. He tears down the image that she desperately, but tenuously clings to. After the explosive confrontation, Stanley exerts his physical dominance over Blanche when he rapes her at the close of the scene. By doing so, Stanley is robbing Blanche of any remaining dignity and control she has.
This act seemingly breaks the spell that Blanche has cast over her sister, as Stella refuses to believe that Stanley assaulted her. Furthermore, Stella, who's connection with Stanley has been renewed by the arrival of their child, allows Stanley to remove Blanche from the home and send her to an institution. The battle of wills that raged between Blanche and Stanley finally ends.

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.