I'll let him speak for his reasons of using a capacitor.
But my 2 cents is that some people feel that the long wire run through the fuse to the trunk has enough voltage drop to be noticed during peak amp conditions (ie: when the bass hits), especially when using low ohm speaker setups. The theory is that the capacitor is supposed to absorb the voltage sags and make the audio cleaner. I'm not going to get involved in whether our ears can hear such things. Nor am I going to remind people that most high-end amps already contain large capacitors anyways. I'm just saying what I know.
This technique goes all the way back to the '80s and '90s. I remember building diode isolators for those systems to eliminate any backfeed into the car from the capacitor so the amp got full power. Again, I can't say if anybody could actually hear the difference, I just helped a guy out who was designing and installing high-end systems at the time. A hobby/business in that I am no longer involved. FWIW, large capacitors were very expensive then.
In this particular case, since everything is in parallel (assuming it is wired properly), if the amp shuts off after the cap is discharged, then the fuse feeding the cap/amp from the battery is blown or the connection is poor. That is the only logical explanation. The capacitor is likely charging via backfeed through the amp somehow but once music is playing, it's not being fed by the battery at all. So it quickly discharges.
A capacitor has the ability to draw a lot of power when discharged (and first connected). It's a dead short until it approaches a charges state. Unlike a battery, a capacitor tries to charge instantly, possibly drawing hundreds of amps. Unless there is built-in inrush circuitry. If there is such circuitry, I'd say something is defective. Or there is some in-rush detection problem with it.
What gauge wire is going to the trunk? Are you sure the fuse is not blown?
Disconnect everything from the power lead coming from the battery including the cap and amp, in the trunk. Use a test light to draw power from the lead. Not a multimeter. It needs to have some load on it. Using an old headlight bulb would be even better to test whether it can draw more than a few milliamps. You can buy a test light at Harbor Freight. You can get an old fashioned headlight bulb from a junkyard. Good luck.