If you stomp with a bit of force, the hammers of keys all across the keyboard strike the strings and make a silly noise. Children love to do it, and in my experience, are then scolded because "it's bad for the piano". I cannot find an authority that this is. So is it actually bad for the piano?
In an upright piano the left pedal (fake una corda) functions by pushing all the hammers closer to the strings. Shortening the hammer travel distance makes the notes play a bit softer. The hammers, in rest position, lie back on the "hammer rest rail", and when you push the left pedal there is a rod that travels up the left side of the piano from the pedal to the hammer rest rail that pushes the rest rail up and the hammers forward. When you stomp on the left pedal, it pushes the hammers forward so quickly that their momentum carries them forward enough to lightly tap the strings. So you can get cool effects by holding down the damper (right) pedal and stomping on the left pedal.
The primary danger of doing this is that sometimes the hammer rest rail will have so much momentum that it jumps off of its attachment to the pedal rod. This will result in the left pedal not working. But the fix for this is extremely simple: just open the piano up and put the left pedal rod back in its proper hole. (A piano technician can do this in 30 seconds, and a handy person could probably figure it out in a few minutes; the hardest part is getting the piano open.)
Other less-likely dangers are that on old or poorly built pianos you could do some damage in the attachment of the bottom board to the piano, or simply break the pedal off. A loose bottom board would negatively affect all 3 pedals, and a broken pedal would be somewhat expensive to replace. But I think the risk of these things happening is low on well-built pianos with children (not adults) stomping on the pedals. When I was growing up we did it all the time without any problems.
There are a few inaccuracies in a previous reply to this question. First, the responder confuses the upright's left "una corda" pedal with the middle "practice mute" pedal on some uprights, which does indeed drop a layer of felt between the hammers and strings. Also, letting kids touch the piano strings (especially the copper-wound bass strings) is not a good idea, as their finger oils will start corroding the strings. Also, letting your kids reach inside the piano and play with the strings is much more likely to result in broken parts (eg. hammer shanks) than letting them stomp on the soft pedal IMO.
It is "bad" for the piano in that a piano is not designed to have the una corda pedal (that's the "soft pedal") stomped on. The danger here is that by stomping, you can damage the mechanism that moves all of the hammers over, thereby breaking the instrument. On upright pianos the una corda pedal is replaced with a practice pedal in which a felt curtain is brought down. Jamming on this felt curtain can break that mechanism as well.
That said, there are specific compositional effects used in contemporary music that call for pedal stomping on both the piano and the harp. If stomping is employed, it needs to be done tastefully and respectfully - not the sort of motions a young child is capable of. Those effects are subtle and are used sparingly, I might add. For the piano specifically, stomping is pretty much confined to the sustain pedal.
There are other, safer ways for them to experiment with a piano without hurting it. For example, holding down the sustain pedal while running fingers or coins across the strings. Another cool effect is striking the string with yarn mallets, or muting strings with your fingers while playing the keys. All of these are cool sounds which do not hurt the instrument in their employment.
Well, it's sort of like a constant expected amount of damage: the more expensive the piano is, the less likely it is to get damaged (due to its robust workmanship), and the less likely you are to want to take chances.
Sort of like using a wrench as a hammer.