I couldn't find anything on YouTube or Google. I know in my Matrix that I needed to remove the trim around the stereo. Once I removed that, I pulled the center knob on the HVAC controls straight out. One screw there, then two screws on the top behind the stereo trim, everything else clips in. Your car should be similar, but work slow and if you find yourself starting to pry hard with no luck, stop and think about what else could be holding it down.
If nobody here has done it on a 2003 - 2008 Corolla, another option is Toyota's TIS website and a two-day subscription.