The inverse of a number is the number that you multiply it by to get 1. In a finite number field there are no fractions, just integers (reduced by the modulus). In the field of numbers mod 23, we ignore multiples of 23. So the inverse of 4 is going to be an integer x (between 1 and 22) such that 4x is equal to 1 (plus a multiple of 23). Since 4×6 = 24 = 1 + 23, it follows that 6 is the inverse of 4 in that number field.