Whats the relationship between capacitors' ability to filter high frequencies and i=C dv/dt
A filter cannot rely on a capacitor alone. It has to work in conjunction with another component, usually a resistor for example. The impedance of a capacitor is related to the formula I = C dv/dt in that if dv/dt becomes larger (a larger change in voltage with respect to time) then the current also becomes larger. 

Given that a sine wave of a higher frequency has a larger dv/dt compared to a a sine wave of a lower frequency, you can probably recognize that the current increases for a larger frequency.

If a resistor and capacitor are used to form a potential divider like this: -



You might be able to recognize that for higher frequencies, more current flows into the capacitor than for lower frequencies. This means that the resistor "drops" more signal voltage at higher frequencies due to that increased current flow and, this means that the output signal amplitude is smaller when higher frequencies are presented to the input.

This circuit is a low-pass filter but if you swap the positions of R and C you get a high-pass filter.