Here is another "functional explanation" of this phenomenon (see also my RG question).
General feature. Unlike resistors that directly decrease current, both capacitors and inductors do it by subtracting their voltage from the source voltage creating this current. You can think of them as varying through time "rechargeable batteries" that oppose their voltage to the initial voltage. For this purpose, these "batteries" are oppositely connected in series to the voltage source. Figuratively speaking, they behave as "ungrateful elements" which "steal" voltage from the source and then use this voltage to resist the source:)
The difference between the two elements is that the capacitor creates an increasing through time voltage opposition (reactance) while the inductor creates a decreasing through time voltage opposition.
The capacitor is "lazy" - so when the input voltage "jumps", the capacitor does not react in the first moment and the current is unaffected. Then it begins gradually increasing its voltage… the current decreases… and, after a long time, the capacitor becomes a voltage source "producing" an equivalent "anti voltage" contrary to the input voltage change... like the op-amp in the circuit of a voltage follower...
The inductor, in contrast, immediately converts the voltage change to an equivalent "anti voltage" and applies it contrary to the input voltage change. Figuratively speaking, the voltage "produced" by the inductor "jumps" with a magnitude equal to the input voltage change. Thus we have two voltage sources ("original" and "cloned") in series that neutralize each other; as a result, the total (effective) voltage and accordingly the current do not change. Over time this opposition decreases and finally, the voltage across the inductor becomes zero again (it behaves as a piece of wire), and the current increases to the new value.