I disagree with this interpretation, but it is a common one. What they are really talking about is quantum mechanics in which particles are not what you think of them as. I.e. you think of particles as solid little BBs (marbles, balls, whatever). In actuality, they are more like putty...spread over an area (really volume).
There maybe a spot where there is "a lot" of matter (this is what you would recognize as the classical, "BB" particle). However, the particle actually extends infinitely out into space...it's just as you get far away from the center, there is virtually nothing.
Now, the reason I don't like saying all particles are waves, is that this is an incorrect interpretation of quantum mechanics and you're not going to understand it (sorry). A particle can be described by a wave function and that wave function represents a probability distribution where the amplitude (like waves have amplitudes) represents the probability of finding the particle at that position.
I don't particularly like the idea of teaching quantum mechanics in high school, frankly. So the truth is, it's one of those things that you are not capable of understanding. So you have two choices: 1) accept it as a factoid about our current understanding of nature or 2) study physics in college and attempt to understand it.
Edit:
And people might point you to the famous double slit experiment which appears to show that the electron apparently behaves like a wave. Indeed the electron creates the same pattern you would expect from a wave (from light). However, it's mathematics. The probability distribution, mathematically, looks like a wave and so it's this probability distribution that is causing the diffraction pattern akin to a light wave.