If you were a mathematician you would prove this using a “continuous deformation”.
A big loop is nearly the same as a slightly smaller loop, is nearly the same as a slightly smaller loop, is nearly the same… If there’s a singularity in the way the loop that goes around it can’t be smoothly deformed into a loop that doesn’t.
He found that he suddenly had a second set of solutions which imply the existence of anti-particles.
At the time the idea of an anti-particle was ridiculous, but a few years later positrons (anti-electrons) were discovered and Dirac was shown to be right.
All of it ends up in the same tiny singularity moments later, but it’s still interesting.