This question used to bother me too when I was in 11th and studied Permutation & Combination for the first time.

See, that's how my Maths teacher explained to me :

In mathematics, all theories are governed by a set of rules and patterns that must be obeyed throughout the entire mathematical system.

Take an example of multiplying a number with itself :

3^5 = 243

3^4 = 81 (243 divided by 3 = 81)

3^3 = 27 (81 divided by 3 = 27)

3^2 = 9 (27 divided by 3 = 9)

3^1 = 3 (9 divided by 3 = 3)

3^0 = 1 (3 divided by 3 =1)

See, a pattern of dividing by 3 is being followed in the above process.

Now, coming to your main answer, we will try to follow a similar approach for finding the factorial :

5! = 120

4! = 24 (120 divided by 5 = 24)

3! = 6 (24 divided by 4 = 6)

2! = 2 (6 divided by 3 = 2)

1! = 1 (2 divided by 2 = 1)

By following the above pattern, we reach to the conclusion :

0! = 1 (1 divided by 1 = 1)

Factorial of n is n! and the value of n! is n!=n×(n−1)×(n−2)×………×1

The value of n! from the above can be also written as n×(n−1)! ⇒n!=n×(n−1)!

Considering the value of n equal to 1, ⇒1!=1!×(1−1)! ⇒1!=1!×(0)! For above condition to be true The value of 0! must be equal to 1.(as 1! = 1)