The construction "noun à verb" signifies that noun is used to perform the action denoted by verb. This is the use you cited. For example:

Machine à laver.

It can also mean that verb is intended to be performed with noun as direct object.

J’ai des choses à dire. = Il y a des choses que je dois dire.

As-tu des vêtements à laver? = As-tu des vêtements qu’il faut laver?

I think this is actually a question about English rather than French.

In English, “to” is used for a lot of different uses. While it is used to build infinitives, like “to play”, I do not believe that

I am going to play.

is “I am going” followed by a “to” infinitive, but rather that the verb is “to be going to”, followed by another verb, without its “to” particle (but I may be wrong on this).

This is where I link with French, as “à” in “cartes à jouer” has nothing to do with infinitives, but could indeed be roughly translated by “to”, as in “cards to play” (used in order to play — hey, another to!).

In effect, the English language renders the expression by “playing cards”, which is quite equivalent to “cards to play” (but again, has nothing to do with infinitives in either of the two languages).

It is to be understand like "Playing cards" that designated the cards themselves rather than the action of playing the cards.

The "préposition à" is used to explain the destination (or goal) of the noun it follows. So "carte à jouer" means "cards that are used for playing"

In English you can be playing cards with playing cards. (A bit redundant of course) and in French "On peut jouer aux cartes avec des cartes à jouer".

Other examples includes:

Fer à repasser

Métier à tisser