There are two disputed theories on why B&W for this film
The movie was supposed to be modeled after documentaries or news coverage of the day which were often in black and white typically. The artistic vision was to create a mockumentary of a day in the life of the band. While the film itself was fantastical and funny, they wanted to film it in a style that was more cinema verite.
The film was said to be greenlit to get the rights to the soundtrack itself with the film thought of as an afterthought. As such, the budget of roughly $500,000 USD (considered low budget even in 1964) was a welcome price to pay to secure rights to the soundtrack. Film historian, Stephen Glynn said the movie was "a low-budget exploitation movie to milk the latest brief musical craze for all it was worth."
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Either way, the black and white gives it a timeless quality and likely also saved United Artists lots of money resulting in a classic movie.