Because this is the first position where the probability of two people in line sharing a birthday is greater than 50%.

To illustrate let's enumerate the first several people in line.
1st person: he's screwed since he is first...
2nd person: There is a 1/365 chance of a shared birthday
3rd person: There are now three combinations that result in a shared birthday, so 3/365.
4th person: now there are 6 combinations leading to a 6/365 chance that there is a shared birthday.

By now you should see the trend that is the probability of there being a shared birthday somewhere in the line at position n is given by
P(shared b-day)_n = Sum(i from 1 to n-1) of i/365

Solving this for n you find the optimal place is n=20. Wait any longer and the odds are greater than 50% that someone will claim the prize before you. Get in line any earlier and you will not have maximized your potential to win.

Of course, if your B-day is February 29 then you are just SOL.