When you stand in a queue, whether it is at a food counter, to pay a bill, or buy a movie ticket, naturally the person who is ahead of you will get service first, and the person at the head of the queue will obviously be the first person to get that service.
The proverb 'First come, first served' thus means that whoever reaches first/is first in a queue will be served/attended first. It need not, though, refer to an actual queue as such but just being the first to arrive at a place, the first to get something, etc.
1. As soon as the car came to a halt, Aparna leaped out and ran to the ice-cream vendor to buy the famed chocolate cone ice-cream, as there were already many customers mingling around. By the time her cousin got out of the car that particular flavour was sold out so Aparna teased her saying "You should have got out faster. First come, first served!"
2. The tickets to the concert were free and would be given to the public on a first come first served basis.
3. It was only when Suresh and his friends arrived at the cricket venue that they realized that the seating arrangement was the same for all and the best seats were taken as it was on a first come first served basis.