I think the OP needs a simple, clear and intuitive explanation of the two electrical arrangements - a charged capacitor connected in series (coupling capacitor) and in parallel (bypass capacitor) to another voltage source.
In such cases, I even suggest to think of the charged capacitor of as a "rechargeable battery"... as a simpler electrical analogy. It is well charged and keeps up its constant voltage.
Coupling capacitors
In the OP's circuit, the input coupling capacitor C3 is charged to the bias voltage set by the R1-R2 voltage divider (I hope OP has some idea what it is)... and is connected in series to the AC input voltage source. So, its (bias) voltage adds to ("shifts up") the bipolar input voltage variations... what the transistor needs.
Similarly, the output coupling capacitor C2 is charged to the output bias (quiescent) voltage... and is connected in series to the output collector voltage. But here, its voltage is subtracted from ("shifts down") the output collector voltage.
So, in both cases, charged "coupling" capacitors (with a constant voltage) are connected in series to voltage sources with varying (AC) voltage. The only difference is their polarity.
Bypass capacitors
In the OP's circuit, the blocking capacitor C3 is connected in parallel to the emitter resistor thus "copying" the voltage across it (I will not discuss what this voltage is). So, it keeps this (its) voltage constant when it tries to change. Thus it fixes the emitter voltage.
Generalization
So, both coupling and blocking capacitors are the same - a charged capacitor acting as a constant voltage source. But in the first case it is connected in series while in the second - in parallel to another voltage source.
And both coupling and blocking capacitors do the same - they keep the voltage across themselves constant. Only, in the first case, they transfer the voltage variations while, in the second case, they "kill" them.
Analogies
A shock absorber is a very good mechanical analogy of the capacitor:
When acting as a "coupling capacitor", it transfers the movement (e.g., of a spring) of the one end to the other.
When acting as a "bypass capacitor" (connected "in parallel" to a spring), it blocks the movements of its ends relative to each other (turns into a hard rod).