Why is Thevenin's theorem useful?

Hint: Thevenin’s Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load. When we get complex circuits with variable resistances (like a rheostat or a resistance which can be switched on and off), Thevenin’s theorem is applied.

Complete answer:
Thevenin's Theorem states that it is possible to simplify any given linear circuit, no matter its complexity, to an equivalent circuit with just one single voltage source and series resistance connected to a load.
The qualification of being ‘linear’ is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). When dealing with passive components (such as resistors, inductors and capacitors), the resulting resistance is always linear. However, there are some components (certain semiconductor components like IC’s) which are nonlinear: that is, their resistance changes with voltage and/or current. Thus, circuits containing these types of components are called nonlinear circuits.
We already have four methods of analysis at our disposal (Branch Current, Mesh Current, Millman's Theorem, and Superposition Theorem) to use in determining voltage across a circuit and current through a circuit, but each of these methods are time-consuming.
Imagine repeating any of these methods over and over for different values of a resistance (changing load resistance is very common in power systems, as multiple loads get switched on and off as needed. The total resistance of their parallel connections keep changing depending on how many are connected at a time). In competitive exams, where each second counts, this could become very time consuming.
Therefore, Thevenin's Theorem comes to our rescue by temporarily removing the load resistance from the original circuit and reducing what's left to an equivalent circuit composed of a single voltage source and series resistance. Thereafter, the load resistance is connected and different values are put in to determine the values of Voltage and Current.

Note:
Thevenin’s Theorem is applicable only for DC circuits. This is because a DC circuit involves constant potential difference and constant current. But, while dealing with AC circuits, the potential difference and the current is constantly changing. Thus, it becomes invalid for AC circuits.