How many members should at least be there in a Club so that it is guaranteed that at least two members have the same month of birth?

This problem is a direct application of the Pigeonhole Principle.
The number of unique birth months in a year is $12$, which represent our "pigeonholes" ($N=12$). To guarantee that at least two members (the "pigeons") must share the same birth month, the number of members must exceed the total number of available categories by at least $1$.

Using the formula:

$$\text{Minimum members required}=N+1$$

$$\text{Minimum members required}=12+1=13$$

If there were only $12$ members, it is theoretically possible for every single individual to be born in a completely unique month. A $13\text{th}$ member absolute breaks this possibility, guaranteeing a matching month.