Exactly the same as what happens to the water in the city reservoir when your tap is not turned on enough to take it all..
It stays where it is.
There is still the same pressure (equivalent to electrical voltage) but no flow (=current, whether liquid or electrical).
It's no different to how the lamps in your home only take so much current, an amp or even a fraction of an amp, even though the house electrical system can supply eg. 100 - 200 Amps continuously.
The voltage is fixed so the power flow through any lamp depends on how much current it allows through (its resistance).
As long as the supply voltage is correct (or within the allowed range for some device), anything designed to work on that voltage will only take the current it needs, no matter how high the current rating of the supply is.
Too high a voltage is what causes damage, like too high a pressure. Too much current (flow) is forced through something and causes damage, making things melt or burst..