Yeah, that is definitely not the correct way to splice coax. The problem with using the aluminum tape is that the adhesive is non-conductive (unless you buy tape that specifically has conductive adhesive, even then it's not a good idea) so there isn't any way for the tape to actually become part of the shielding. It'd be like trying to patch a water hose using electrical tape. Sure, it might look like it'll work, right up until you turn on the water and let it build pressure, then it will just leak right past the tape. And in this case it's like there is water on both the inside and outside of the hose, so it can get in and out.
And as RadioMike said, it's unlikely that the splice will actually match the impedance of the original coax, which does make a difference in your ability to send out and receive signals. The drastic reduction in distance and isolation between the center conductor and the shielding is also a real problem, as it could effectively create a "short" at radio frequencies, since RF voltages don't behave the same way a DC voltage applied to the cable would. This could also be what is causing the noise leakage into the coax.
If you really have to splice the cable back together, then you need something like this (but rated for the correct impedance):
http://www.homedepot.com/p/Tyco-Electronics-Gel-Coax-Splice-Kit-1-Clam-CPGI-569224-1-K/202204333
Notice how you're putting connectors on the cable ends and using a male-male connector to hook them back together. This ensures the shielding and insulation is consistent the whole way around the center conductor. Even having a small nick in the shielding is enough to let noise in/out dependinbg on the frequency and signal strength.
Really though, like RadioMike said, you're better off just getting a new cable and replacing the whole run. I'd make your friend pay for it and then use the old cable as a whip to teach him to never cut cables/wires all willy-nilly.
This is a problem I have with the CB radio crowd, far too many of them have absolutely no idea what they are doing, or how any of it even works, so you end up with things like that terrible splicing example, or modified radios that transmit out of band, over power limits, and with splatter and noise for everyone within 5 harmonics. The worst part is a lot of these people just don't seem to give a dang about it. Such a stereotypical "manly man" trucker mentality. "My way or the highway bud, I've been doing it this way for years and it's never been a problem for me so screw you."
Thank you H3Wolf for being willing to admit you don't know something and actually ask for help from people who do know what they are talking about. If more people were like that you wouldn't see stuff like that "splicing" page.