It seems to derive from a structure like "la victoire/le prix revient au premier qui…" with "au premier" then focused by a c’est cleft:

Ils avaient inventé un jeu … C’était au premier qui reconnaîtrait l’auteur de ces clichés [que reviendrait la victoire]

Dans la « course à l’espace » entre les États-Unis et l’Union soviétique, c’est au premier qui atteindra la Lune [que reviendra la victoire].

Nous nous sommes précipités hors du bateau et c’était au premier qui trouvait les toilettes [que reviendrait le prix (de se soulager en premier)]

L’après-midi à la sieste, c’était au premier qui me demanderait s’il pouvait dormir avec Calinours [que reviendrait la récompense (de dormir avec)]

The general condition to use that sentence is that the first (au premier), the fastest (au plus rapide), or generally the best at something (à la plus maline, aux plus fourbes, etc.) will gain an advantage, a recompense, or a victory.

The expression "To the victor go the spoils" is quite similar in spirit, I think.

It’s hard to find a solid English translation that will fit in all contexts, but a template like this should work:

The [Xst] to [do action] will win / enjoy the prize / get an advantage over the others

The first to recognize those clichés’ author would win

The first to reach the moon shall win

The first to find the toilet would enjoy the prize (to relieve themselve first)

The first to ask me to sleep with Calinours would enjoy the prize (to sleep with the plushie)

“C’est au premier qui…” indicates that there is a competition of some sort, and that the goal of the competition is to be the first to do something. This is an example of a more general construction which indicates the goal of a competition: “C’est à qui …” or “C’est à celui qui …”, sometimes with a qualifier stating that the winner has to be the first or last to reach the goal.

C’est au premier qui atteindra la Lune.
C’est à qui atteindra la Lune le premier.   (equivalent to the previous one)
C’est au dernier qui freinera.
C’est à qui freinera le dernier.   (equivalent to the previous one)
C’est à qui lancera la balle le plus loin.
C’est à qui criera le plus fort.

This construction can be used even if no competition has been mentioned explicitly, since it conveys the idea of a competition. For example:

Nous nous sommes précipités hors du bateau et c’était au premier qui trouvait les toilettes.

An adequate translation in English would be “we all ran off the boat and raced for the toilets”. “C’était au premier qui …” implies that there is a competition to find (and implicitly reach) the toilets first, i.e. there’s a race.

If the idea of a competition is already established, as in your space race example, then this phrase merely establishes what the goal is.