The "a + infinitive" construction is a way to ask for a given action to be done. You can read about this construction if paragraphs 42.5p and 42.5q of the Spanish Gramática. There it is said that those expressions belong to the colloquial Spanish. Many of the examples of this construction are characterized by their expeditious tone:
A callar y a tus cosas, metomentodo.
¡A ordeñar las ovejas, ya!
¡A callar todos!
Nonetheless, they are also used in contexts where something is just recommended:
A dormir, mujer. Mañana lo acompañaremos al tren.
They are also used in vehement invitations to perform some activity, often pleasant:
¡A bailar, a bailar!
¡A vivir que son dos días!
¡A disfrutar se ha ido, a pasarlo bien!
Sometimes the invitation is not pleasant at all:
¡A jorobarse tocan y punto en boca!
You can even try this as a farewell in a colloquial letter or conversation:
A seguir bien.
I would say that these constructions often come from more elaborated sentences where an initial verb is omitted, although it's not always easy to know what verb has been omitted (if it exists at all):
¡Venid a comer! ¡La sopa está lista!
¡Ponte a ordeñar las ovejas, ya!
¡Vamos a vivir, que son dos días!
In your example, a comer can be translated as time to play because of the lunch times, but you cannot always use the "time to..." construction to translate. In the "a ordeñar las ovejas, ya" example I would translate that as "go milk the sheep", the "a callar" would be something like "shut up", as an order. You have to undertand the context and what the speaker really meant in order to give a proper translation for the sentence.