When a string is plucked near the end where it is attached, the standing wave kind of makes jerking motions. It's just like when you jerk one end of a rope that is attached somewhere. You send a pulse up the rope which comes down again. If you look at one point on the rope, it might be making a quick movement one way, with a slower recovery in the other. This is where we get the sawtooth.

For a sawtooth with all the harmonic, we have to use a bow. A bow directly stimulates a strong sawtooth movement simply because the string adheres to it and releases, many times a second. This will tune itself to a resonant frequency, along with the first bunch of its harmonics.