Okay... Let me see if I can explain this. (Lou can decide if this makes any sense. And everyone else can let me know if something is not accurate somewhere.)
Tap water is intended for people to drink. So water companies add chlorine to public drinking water as a purifying agent to kill any bacteria that might be present. Otherwise, it could make us very sick. (This is the same reason we add chlorine to swimming pools.) One drawback to using chlorine, though, is that it dissipates (or degasses) very fast. This happens naturally in about 24 hours. With the chlorine gone a day later, new bacteria can start to grow now. So that water wouldn't be very safe for us to drink anymore.
Water companies know this, of course, so to prolong this margin of safety for the tap water humans consume, they also add chloramine. Chloramine is an agent that binds chlorine with ammonia at a molecular level. Once the added chlorine has degassed and is already gone, the chloramine will still be present for a lot longer period of time (because it's bound with ammonia and can't degass right away). This chloramine then continues to kill or retard bacterial growth.
Fish, on the other hand, can't live for very long in a glass box (or bowl) full of sterilized/drinking water because this water no longers supports the living organisms they need (like bacteria) to survive there. Tap water that hasn't been conditioned with something like Prime to remove the chlorine/chloramine is going to kill all types of bacteria indiscriminantly. Not only the "bad" bacteria that makes us sick, but also the "good" bacteria we're trying to establish in our fish tanks. (This is where that nitrogen cycle comes in.) This is also why we're not supposed to rinse filter media under the faucet. Plain tap water will kill the beneficial bacteria growing on it. Why? Because it contains chlorine and chloramine which was added to do exactly that - kill bacteria. The other factor we need to know is that while chlorine and chloramine are harmless for people, they are both very toxic to fish.
So now, when we add Prime (or some other dechlor/conditioner) to the tap water, several things happen at the same time.
1. It gets rid of chlorine right away. So that's no longer toxic to fish or bacteria and we don't have to wait 24 hrs for it to degas on its own.
2. It neutralizes chloramine in a way that breaks down the bond between chlorine and ammonia. This frees the chlorine again... (refer back to Item 1 to see where that goes).
3. It alters the remaining ammonia somehow so it (1) will still feed the bacteria colony which converts it to nitrite..., and (2) it makes it NON-toxic to fish.
We come along then with our trusty API test kit to take another reading. The ammonia test detects a higher level of ammonia than we had before we did the water change. So we wonder..
"Hey! Where did all that ammonia come from that wasn't there before I did the water change?"
(This is the good part.)
Answer: It came from the chloramine our water district adds to the tap water we just used to refill the fish tank.
So, even though there's technically more ammonia present in the water than there was before, we can just ignore it because it's the "safe" kind now. And we consider this a "false positive" ammonia test result. That's because the ammonia has been "conditioned" by Prime (see Item 3 above) so it's now in a different form that's no longer toxic to fish.
All is well. Are we done now? Well, not quite yet. There's still another question to answer...
"Okay. So, what happens if we refill the tank without using a conditioning treatment (like Prime) for the tap water we add?"
If we happen to be filtering the water with (fresh) activated carbon, that will pull out all the chlorine... great! (See Item 1.) It will also break down the chloramine bond (chlorine + ammonia).. yay! (See Item 2.) But as the chlorine gets removed, this will leave the ammonia behind... unaltered. (So much for Item 3.)
For us to get rid of the ammonia we've just introduced, we rely on the bacteria colony living there to process it for us and convert it to nitrites > nitrates. But in the process of everything else we've just done, the untreated chlorine and chloramine we added (in the tap water) have greatly diminished the beneficial bacteria we cycled the tank to establish... before the carbon gets a chance to remove it for us. If the cycle no longer functions at it's previous level, the fish are at risk from the high ammonia spike we just caused and our tank will go through a mini-cycle to recover again.