How does one explain that the electrons that we know as matter behave like waves?

I think what you meant to say is: "how do you explain that electrons which we know as particles behave like waves." It's a consequence of the dual nature of matter - that matter behaves as both particles and waves - which is at the heart of quantum mechanics. The classic dual slit experiment shows the weirdness of quantum mechanics quite well: when you shoot electrons at a dual slit the result is a diffraction pattern consistent with waves. One way to think of this is as an application of the Heisenburg uncertanty principle - if you shoot an electron at the dual slit screen at a known velocity, you can't pin down the electron's position because of the uncertainty principle. Consequently its position isn't exact, but rather is a probability distribution, and the electron behaves as if it is in all places at the same time in accordance with that probability distribution. The result is the electron behaves as if it were a wave function.

Here is a pretty good description of the wave-particle duality of nature: Wave?particle duality - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality)