Not an expert, but my understanding is that the multiverse (at least, what we today associate as the multiverse) came about due to the many worlds interpretation of quantum mechanics. Basically, quantum physicists had an observation - particles were moving as though they were being pushed by an invisible wave, and they would pick a random position based on that wave when observed.

The most prevalent explanation for this behavior is the Copenhagen interpretation, which states that the particle is the invisible wave, and the wave collapses into a particle when it is measured. But another common interpretation is the many worlds interpretation, which states that the invisible wave is just a statistical probability of where the particle is. And the reason why the particle seems to pick a random point on the wave when observed is actually because the particle creates branching timelines, and we can only observe what happens in our own timelines. Hence, it seems random to us.

I speculate that the idea of multiple parallel timelines, each slightly different, was probably pretty popular with scifi writers, especially since it’s an easy way to portray “what if” scenarios in their stories, and so the concept became popular because of that