Mains supplies do not limit the current. If you overload a mains supply, the breaker or fuse will blow.
However, the mains does not change instantaneously. The rate of change is well defined. Mains voltage is a sinewave, and the rate of change of a sinewave is also a sinewave (or cosine wave if you want to be exact). The peak value of the rate of change is proportional to the voltage and the frequency. The current in a capacitor is proportional to the rate of change of voltage and the capacitance.
So the current in a capacitor that is connected to a mains supply is proportional to the voltage, the frequency and the capacitance.
An hydraulic analogy would be the tidal flow into or out of an estuary. The flow rate is proportional to the range of the tide (analogous to voltage), the frequency of the tide, and the area of the estuary (analogous to the capacitance). If the tide were to go in and out twice as often, twice as much water would have to flow in and out, which is like the current being twice as much.