In-universe, the TARDIS' chameleon circuit got jammed while it was disguised as a police box, a common enough sight in London in 1963.
The writers needed something that would be able to go unnoticed in London at the time. The original plan was it as invisible (an option that is used occasionally, as in "The Impossible Astronaut"), but they decided to "hide it in plain sight" instead, by disguising it.
According to Peter Bryant (one of the early producers, although he wasn't around at the very start of the show), the plan was for it to be disguised in each story, but they quickly realized that would be too expensive. If you want to set a story in Regency London, the BBC already has all the props, costumes, and sets you need—unless you need a piano with a door in that the actors can comfortably step through.
The police box has dated, of course -- but in a way that benefits the show. In 1963, anyone seeing the prop would think "police box". Nowadays, and for decades past, anyone seeing the prop -- or a real police box in a museum, for that matter -- immediately thinks "TARDIS".