Well, I'm just echoing what other people have said. I wish I had read this forum before I broke mine.
When you do the repair, you'll be able to tell why this happens. The top stack of digits is turned by the input gear from the speedometer. A gear on the far right of the stack turns a gear in-between the top and bottom stacks. The middle gear then turns a gear on the far right of the bottom stack. Follow all that?
The only thing keeping the top gear fastened to the shaft is friction. If something causes it to keep still while the shaft is being driven, it can break loose and just spin freely on the shaft. Usually doesn't happen until the car has a lot of miles on it (say 100K or so).
When you reset the trip meter, it binds up the gear that is driven by the top stack of digits. Since the input shaft is still trying to turn, but the gear that drives the trip meter cannot turn, something has to give and the shaft starts slipping inside the silver gear.
Probably not as good an explanation as seeing it for yourself. When you fix it, it will all make sense.