At first, one needs to note that there's no single role of "particle" or "wave" in micro-world. Everything at micro-scale acts according to wave-particle duality principle. Hence, the De_Broglie wave-length of any object which has a momentum. Trivial example, light - is it a wave or particle ? None of them, but both. Light interacts with matter (is absorbed, emitted, scattered) as photon particle, but it propagates in space as a wave - hence the diffraction or interference pattern you see on screen.
Same thing applies to an electron. It interacts with detector screen as a particle, but travels through slits as some sort of wave, because otherwise you would NOT see an interference pattern on electron detector. Only waves can interfere with themselves. Now, you may ask what kind of wave is that ? What we know is that wave function modulus squared is probability density of finding electron at given detector place. But there is no "deeper" understanding of wave nature of electron.
Of course it would be silly to imagine that electron somehow "splits in half" when going though both slits at once and then joins back into complete form after slits. But I guess, neither a photon is divided like that too due to slits. Particle integrity must be obeyed, because if we would imagine that electron is broken into two halfs at slits - then each one would have $0.5e$ elementary charge, which would be a complete nonsense.
