This one is hard to dispute per say since it's difficult to assess exactly what is meant by "speed up" your computer. But I'll attempt to tackle it anyway.
Aero has little impact on the main processor(s) since it is leveraging the GPU to do the heavy lifting though a mechanism called the Desktop Window Manager. This layer in Windows Vista handles the actual drawing to the screen, much in a similar fashion as GDI, but does it through what can be referred to as compositing. Essentially what happens is that, when DWM is enabled, everything drawn to the screen is drawn to an in memory location and then quickly drawn to the screen. You can think of this as a double buffering drawing technique, but as I said, the heavy lifting is done via the GPU.
The immediate thing that you gain from this is the removal of the "screen tearing" associated with moving windows around on the screen. As an experiment on Windows XP, have a lot of icons on your desktop. Then take a good sized window and drag it around on the screen. Notice how the icons underneath lag a little bit in their redrawing. Also notice how the window seems to ghost just a little. Now try the same thing on a DWM enabled desktop.
So with this little experiement, how exactly does turnng it off "speed up" your computer? There is an immediate appearance penalty for turning it off (other than the loss of the glass effect) so I'm not sure I see the benefit, or for that matter, any performance gain for turning it off.
Update: Don't just take my word for it. ;-) Read here for someone elses "in-depth" study.