When the hold down timer starts, it is as if the router is putting its hands over its ears, and mumbling "I am not going to believe any new updates regarding the network that is in hold down". The reason for the hold down is to minimize the impact from routers who perhaps didn't get the message that the network (route) was down and are still advertising it with a reachable (but not as good as the original source) hop count. Sort of like a timeout, until the network has had a better chance to convergence. For example, lets say that R3 has been learning about the 126.96.36.199/8 network from a neighbor (not shown), R3 has been advertising that route to R2 with a hop count (cost) of 7. When the hold down timer starts for that 188.8.131.52/8 network, R2 will not believe any new updates from R1 or anyone else (until the hold down timer expires), unless the new metric is 6 or less (better metric).
Here is a text diagram, with R1 - R2 - R3 we can use as a reference.
additional neighbor(s) to R3
Even if the original router who advertised the route comes back on line, and R3 begins advertising the specific route of 184.108.40.206/8 (which was in hold down a moment ago) it will still be in hold down, (until the timer expires), even though R3 is advertising the route with the same metric (hop count) of 7 that it was originally. That is the theory behind the hold down timer. (The network is considered "possibly down" during this time, and the router will still forward based on the routing information it had as it went into hold down. Packet will still flow towards what is believed to be the correct direction.)
With RIP, a neighbor would be a directly connected neighbor. In this example, R2 has only 2 possible RIP neighbors, R1 and R3.
The hold down timer begins, by default, when an update regarding that route hasn't been seen for 180 seconds.
In reality, Cisco modifies code occasionally, and the actual results will depend on the version of IOS's reactions to triggered updates, poisoned routes (poison reverse), etc. For example, currently, when a poisoned route (16 hops) is advertised by a neighbor, the router today will cancel the hold down timer, and begin sharing the bad news (16 hops) to its neighbors.
Here is a link that goes through an entire debug of the process. It may be a bit more than you need, but a good reference.