Ok, assume there are n people on the planet.
1. Each one may have from 0 to n-1 friends (he can't be friend with himself!)
2. If there are NOT at least two persons with the same number of friends, then each person should have a different number of friends. This means that there should be one with 0 friends, one with 1 friend, one with 2 friends and so on, up until the last one who should have n-1 friends.
3. BUT the guy who has n-1 friends (all the other except himself), could NOT be friend with the guy who has 0 friends.

Statement 3 proves that statement 2 is false, therefore there are at least 2 persons with the same number of friends.

This was a rather mathematical solution. To understand it more practically, assume that there are 4 persons in the world, A, B, C and D. Assume also that statement 2 is true. This means that A has 0 friends, B has 1 friend, C has 2 friends and D has 3 friends. If A has no friends, then who is the 3rd friend of D ????