You can look at the problems it solved.
Volume. Concerts got louder, sound systems became more common and complex , with musicians bringing their own amplifiers. An upright acoustic bass at the time had to be miked. At higher SPL's ( sound pressure levels) microphones become rather more difficult to use, because the sound coming out of the main speakers ( or, worse, the monitor speakers on stage) can be picked up by a microphone and re-amplified, creating feedback. A magnetic coil pickup on an electric bass guitar has much less trouble. Today there are sophisticated ways to stop feedback in the processing of the signals in the sound system, but it was more difficult in 1950-60.
Sustain. An upright bass has a beautiful rhythmic pizzicato sound, but it's short- a "tum-tum-tum-tum". An electric bass guitar ( even a Hofner, like Paul McCartney's, that tried to capture some acoustic bass qualities) can lay down long sustained notes as well, for those "power chords".
Portability, durability. Easier to haul around a bass guitar, easier to travel with it.
Accessibility. Someone who starts on guitar can move over to bass guitar- they both have frets. Someone who starts on guitar will have more trouble jumping to acoustic bass, and guitar was more commonly what rock n' rollers started with.
Things have changed. If the upright electric bass had been created in 1950 ( which mostly solved problems 1-3) it's possible the bass guitar would have had a slower acceptance. And fitting an upright acoustic bass with piezo pickups enables them to play in pretty loud concerts these days, and they do, so people who want to capture the sound of Elvis' Bill Black on stage can get someone to play acoustic upright bass and not worry about SPL's.