How do the Jewish food laws affect the Jewish believers?

Answer 1

By the fact that Jews keep kosher.

In order to be kosher, food has to be prepared according to the kosher-laws (see Deuteronomy ch.14). These are explained in detail in the Talmud-volume of Chullin.
* Meat must be from those land animals which have split hooves and chew their cud (such as beef, venison and mutton).
* Fish have to have scales and fins. Shellfish are not kosher.
* Birds cannot be among those listed as forbidden in the Torah (Deuteronomy ch.14) and cannot be hunters/scavengers. In actual practice, today we eat only those species concerning which we have a tradition that they're permitted, such as domestic chicken, geese, pigeons and turkey.

* Animals must be slaughtered in the manner specified by Jewish law and must be free of all disease. In actual practice, those who keep kosher purchase meat which is certified as having been prepared in the kosher manner.
* As much blood as possible must be removed from meat before cooking, since consumption of blood is forbidden (Leviticus ch.17). This is done at home or by the kosher butcher, through salting, soaking and rinsing.

* Dairy and meat cannot be combined in the same meal and there's a waiting period between eating one and then the other. After dairy: 1/2 hour. After meat: 6 hours for most Jewish communities.

* Fruits and vegetables should be checked to be sure they're free of bugs. Some Jews avoid cauliflower, asparagus, and the like, because of the difficulty in checking them.

Additionally, food must be prepared and handled following kashrut-laws and with kosher ingredients only. Any food that does not meet these requirements cannot be eaten by those who are religiously observant. Foods which are purchased should be labeled as having had kosher supervision during their processing.

See also:

Why do some Jews not keep kosher?

Answer 2

They determine what, where and with whom Jews can eat.