This phrase is an ellipsis for: Who comes first is served first. It means somebody who precedes others will have his needs attended to earliest. The idea is old, but the first recorded use of the phrase was in 1545.

Übersetzung:

Wer zuerst kommt, mahlt zuerst.

Gegenteil(e)

The opposite of "first come, first served" is "who is late, has to take the leftovers".

Synonym(e):

You could also say: Who arrives first, has his needs attended first.