Because it has some wave-like properties. The double-slit experiment, which used photons instead of electrons, was *the* experiment to prove “wave-particle duality”, the fact that elementary particles are something that behaves like both a wave and a particle.
A particle obviously can only go through one slit at a time. A wave can obviously go through both (just imagine a single wave on the ocean hitting a seawall with two gaps in it). In the latter case, you’ll get a “piece” of wave coming out of each gap and those two pieces can happily interfere with each other. Electrons are like photons, they have some wave-like and some particle-like properties.
The whole “interferes with itself” thing is the electron showing off it’s wave-like stuff.
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