Q.Why does the gain of an amplifier decreases at very low and very high frequencies?

Different capacitors dominate at the two ends of the frequency range, each reducing the gain. …

[!TLDR]

Low-f: coupling-capacitor reactance high. High-f: shunt-capacitance reactance low. Both cut the gain.

At low frequencies the reactance of the coupling capacitor $X_C = 1/(2\pi f C)$ is large, so an appreciable part of the signal is dropped across it and only a small part reaches the next stage — the gain is small. At high frequencies the inter-electrode and stray (shunt) capacitances have small reactance and act almost like short circuits to ground, diverting the signal away from the load and increasing the loading effect — so the gain again falls. Only in the mid-frequency band are both these effects negligible, giving a constant maximum gain. …