Because you are supposed to be on-board (Read: boarding pass scanned) 15 minutes before departure. After you're within that 15 minute window, you lose rights to your specific seat and it can be given to a standby passenger or a non-revenue passenger. Gate Agents routinely make this announcement when boarding.

There's no guarantee that even if you're at the airport coming in off a delay that you'll make it to the flight by the time the door closes. So, that's why they have to draw the line. Occasionally exceptions are made as pointed out up-thread, but in general the line has to be drawn and 15 minutes is that line.