The main benefit of Thevenin’s theorem is that it helps to simplify a circuit which has got n number of batteries and resistors (and contain two terminals), into a single voltage source and a resistor connected in series with that voltage source, along the two terminals.
Let us now see that the theorem has to say and it states that “The current through a resistor R connected across any two points of a network which contains one or more sources of emf, can be calculated by dividing the potential difference between those two points in the open circuit mode (i.e. when R is removed and the potential difference between the two end of R is taken into account), by the R + r where r stands for the equivalent resistance between the two points when batteries are replaced by their internal resistances".
This theorem is also known by the name of Helmholtz’s theorem since it was also stated by Helmholtz prior to Thevenin but that is more of a matter of historical concern and the implications of the theorem for network analysis remain equally useful regardless of the fact that who put it forth first. It can also be seen that this theorem is an extension of the Superposition theorem, which is one of the most fundamental network theorems in electrical engineering.
If you find the theorem bit difficult to digest, simply remember that provides a way to replace a network having two terminals and source of emf with a simple circuit having one single source of emf and a single resistance connected in series to each other. The simplified circuit which is generated using the circuit is known as the Thevenin’s equivalent circuit of the original circuit.
I will not go into the exact mathematical analysis and simplification of circuits but the diagram shown below will give you an idea about how this theorem can be used to simplify circuits. If you see that diagram, the figure on the left hand side shows and original circuit with two terminals, four resistances and a source of emf. The figure on the right hand side shows the same circuit in the form of Thevenin’s equivalent circuit where all the resistances have been replaced by a single resistance. It must be noted here that although the single source of emf still remains the same, its value in the Thevenin’s equivalent circuit changes as per the description in the theorem and the same can be calculated from the theorem as well.
As with most other theorems this theorem also works well in the linear range of circuits but not in the non-linear range.