10^100 means writing 1 followed by 100 zeroes.
Hence 101 digits are required to represent it in decimal notation.

Another way to think about it is this....imagine you have 100 10's lined up to multiply by each other. Multiply the first 2 and you get 100 which is 3 digits. Now you know that each additional 10 will add one more digit. You have 98 more 10's, so you take the 3 digits you already have and add the 98 additional digits and you get 101.

10^1 = 10 Use 2 digits (1 + 1)
10^2 = 100 Use 3 digits (2 + 1)
10^3 = 1000 Use 4 digits (3 + 1)

=> 10^n would require n+1 digits