Why is it that the amplitude of a probability wave is the sign of "a single particle"?
Take a spring or coil of any kind. Look at it from the side, or even better, project a shadow of it onto a piece of paper. In both cases you will see what looks like a sinusoidal wave complete with peaks, troughs, and zero points. But the spring has a smooth, constant radius that doesn't show any such peaks and troughs.
A quantum amplitude is very much like the radius of a spring, and the sinusoidal representations of a quantum wave function that you see in most text books are very similar to the shadows projected by such springs.
When Strassler said that the amplitude represents "a single particle," he was trying to emphasize that a sequence of peaks and troughs separated by zero points is really a single entity (like the spring) with a single smoothly changing amplitude (the radius of the spring). The peaks and troughs are just illusions acquired by projecting the wave function onto an overly simple 2D screen or page of paper.
(I added the above on 2013-03-04 My original answer, sightly edited, is below.)
I looked at Matt Strassler's blog, and I'm pretty sure his real intent was just to keep the questioner from thinking that a particle is always a single peak in a probability wave function. That is just wrong, and Professor Strassler was trying to make sure that readers didn't get into the habit of thinking in such terms.
Here's a slightly different way of looking at probability wave functions that may help.
In an earlier answer about Fourier transforms I argued that a much better way to think of probability wave functions is to use a complex plane perpendicular to real space, and then visualize how the amplitude or height of the wave function maps into that space. The problem is that such an approach requires thinking in a five-dimensional space, which is tough! However, you can cheat a bit by looking at only one XYZ dimension at a time (e.g. the length X of a box), and "borrowing" YZ to represent the complex plane.
Now, if you do that for say an electron bouncing back and forth between two ends of a box with length along X, the idea of "peaks" and "troughs" in the wave function pretty much disappears. Instead, you get various sorts of moving helical coils (moving electrons) and skip-rope-like stable states (the resonant or "stationary" wave function solutions) along a rope (representing amplitude) that stretches from one end of the X box to the other.
I should mention that it never ceases to amaze me just how close the differential equations that control rotating loops in an ordinary string or rope are to the equations that control this composite real-and-complex representation of wave functions. For example, if you take a hose laying in the yard and give it a quick circular jerk at one end, you will see a short helical wave move from your hand and travel down the length of the rope. What's remarkable is that both the helix and the way it moves have almost exact mappings into the wave function of an electron wave packet moving through space. Moreover, the "skip rope" loop solutions represent the resonant states in which you in effect have the electron helices going both directions at once (a "quantum superposition" of left and right moving states of the same electron).
In this rotating-rope model you only get peaks and troughs when you project a shadow of these coils onto a piece of paper. Think for a moment about how a spring or Slinky looks from the side and you can see how the usual sinusoidal curves with peaks and troughs can emerge from by limiting your perspective two only two dimensions (the projected shadow of the amplitude rope).
I'm pretty sure in fact that that was the message Professor Strassler was trying to get over in his comment: There is just one more-or-less continuous amplitude (the coils of the helix) where the particle is located, with those peaks and trough literally just being shadows of the underlying reality of the wave function.
Incidentally, I have to mention it: The rope analogy becomes even more powerful if you "standardize" the total volume enclosed by the various rotating coils and loops along the X length of the rope. If you do that, then each the volume enclosed along any segment of the rope becomes the probability of finding the particle along that part of the X rope. So, if you have two resonant loops (think of expert skip-ropers with double loops) along X, then the probability of finding the electron becomes 50% in either loop -- and 0% in the center! The electron sort of magically "tunnels through" that part of real space to get to either side. If you look carefully, however, the rope itself is moving like crazy at that same central location, so it's not quite as simple as saying that the electron is "not there" at the center. Its wave function is very active there, but just does not allow the particle to be found at that spot if you look for it.
I should point out also that by intent I just undid my whole argument! That is, while I just argued that the projected peaks and troughs of a wave function do not accurately convey the continuity their underlying complex amplitudes, it is not correct to say that the wave functions amplitudes for a single particle are contiguous. It is actually very common for them not to be, since for example the various lobes ("skip rope loops") of electron orbitals in atoms are examples of discontinuous electron wave function amplitudes. Such electrons can be found in certain disconnected regions of space, but not in the regions in between.
So, bottom line questions and answers:
Is the existence of an amplitude a sign of a particle? Yes, pretty much by definition.
Do the peaks and troughs of the real part of the wave function mean anything about where the particle really is? No; you must use the complex wave function for that.
If you find a single stretch of complex amplitude that surrounded entirely by zero amplitudes in XYZ space, does that stretch necessarily represent an entire particle? No, definitely not, since that very situation happens all the time in atoms. The wave function may be broken up into many pieces, some conceivably quite far apart in XYZ space.