Differentiation reveals the rate-of-change (or instantaneous rate-of-use) of the original quantity or equation.
Integration reveals the cumulative effect of the original quantity or equation.
In the physical world, velocity is the cumulative effect of acceleration (v=u+at), distance covered is the cumulative effect of velocity (s=ut). The first equation covers the case of
constant acceleration, while the concept of integration generalises this for any arbitrary time-varying acceleration. The second equation covers the case of
constant velocity, while the concept of integration generalises this for any arbitrary time-varying velocity.
Current is the movement of charge, so a charge build-up (eg on a capacitor) is the cumulative effect of a current-flow (Q=It), for a constant current. Integration Q = INTGL (I.t) dt covers the general case of arbitrary time-varying current.
Power is the
rate of use of energy - so power is the derivative of energy-flow, or total energy used is the time-integral of power use.
In earlier stages you'll mostly study integration (cumulative effects) over
time, but you can also integrate (sum) over wavelength - eg the total energy output by a lightbulb is the sum (integral) of the energy emitted at each wavelength (colour).
You also commonly intgrate (sum) over space or direction, eg the total light output by a bulb is the sum of the light emitted in each direction (it will not in general be uniform in all directions) - although you're rather less likely to have a nice equation for this one!!!
It's just a natural, reasonable (even obvious) thing when you "get" it.