Some intuition,
Say we have total friends,n = 10
So, at most one people can have n-1 friends = 9 = r
But we are supposed to pick $\bf{at most}$ 1 friend, that means, I can pick 0(no friend) or a 1 friend.
So, two cases,
i. Minimum case:
If I pick 1 friend friend of each out of n = 10 people
then $\frac{n}{r^2+1}$ = 10/(1^2+1) = 5.
ii. Maximum Case:
If 0 friend picked.
then $\frac{n}{r^2+1}$ = 10/(0^2 + 1) = 10.
This calculation can be generalized with the help of a graph.