You can think of particles as wave packets, and in some respects that's largely how they're considered by quantum field theory - essentially a large enough blip in the field becomes an actual particle. Hence also explaining why the particle can diffract with itself: a significant part of it's wave envelope went through the second slit. The extra oddity here is if you put two detectors so you can track which slit the particle went through - then you get two distinct piles and no diffraction.
There is actually another real world analogy that fits some circumstances, oil droplets riding on 'pilot waves', there are some vids of this on youtube and gives an idea how apparently individual particles can diffract with themselves.
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