Wow, cool, thanks for the compliment!
Since it's an Audiovox remote starter, the bypass module is likely (but not necessarily) an Audiovox AS-TCBM. Another popular choice at the time your car was made would have been a 555U by DEI (Directed Electronics).
Anyway, regardless of brand, it's a black box a little bit bigger than a pack of cigarettes. It has a couple of wires going to it, there's a spare key (probably your valet key) inside, and then a "ring" that goes around the keyhole on the car's ignition switch.
The remote starter turns on the bypass module's internal relay, which then connects the ring that's around the keyhole, to another ring (inside the box) that's around the head of the key. The car "sees" a valid key, and then can be remote started.
I couldn't find an install guide online for the ASTCBM. Here's a guide for the newer DEI 556U. (Note that the 556U has a lot of wires that can be connected in various ways; ignore that part for the time being.)
LINK:
http://www.directedstore.com/manuals/556UW.pdf
There are really only four basic possible installation errors: The ring around the keyhole is out of position, the key is out of position inside the box, the box isn't being activated by the remote starter, or the key inside is bad and/or unprogrammed to the car.
My experience is that it's easier for a novice to take down the driver's underdash panel, than it is to take the shroud off the steering wheel. The clips on the shroud are easily broken. With that in mind, I'll start my troubleshooting instructions where I think it'll be easiest for you. If you think removing the shroud is easier than the underdash, you could switch some of these steps around:
1. Obtain access to the black box under the driver's dash where the key is inside. Put your hands on it, and activate the remote starter. You should hear/feel its internal relay click on immediately. If you cancel the remote start (tap the brake), the box should click off after a few seconds.
If it doesn't click, it could be defective, or might not be wired to the remote starter correctly. Post back with results.
2. The box has four screws. Unscrew them, take the box apart, and find the key inside. You'll clearly see where the head of the key sits under a coil of wire. (Look at Page 3 of the link I posted above.)
Take the key out, put it in the ignition, and try the remote start. If the car starts, you know your key is good. If it doesn't, get the key reprogrammed or replaced.
3. My experience is that the letter H from the valet key MUST be positioned right under the coil of wire. The ASTCBM isn't large enough for the oversized Honda key; you may have to break open its casing, and secure the now-protruding key with tape.
Anyway, just plug the thing in (you don't even need the casing), hold the key in that position, and try the remote starter again.
If it works in that position, that was the problem. Secure it permanently in that position somehow.
4. If it's not that, it's the ring that's fallen out of position on the front of the key cylinder. The shroud around the column has three screws on the bottom, and then the sides and behind the steering wheel are easily-broken clips. Maybe a couple are already broken from the original install! Even if you break them all, I think the whole shroud is less than $20 at the dealer. (TIP: Have the car warm inside; cold plastic panels are brittle.)
Anyway, look at Page 4 of that 556U install guide. The drawing does a better explanation than I can in text, but you'll see that the position of the ring is important, especially so on Honda and Toyota. It has to be nice and tight right there around the keyhole.
Put it where you think is good, temporarily, and try the remote start. If it works, use tape or a zip tie to secure it, then put the column back together.