For the purposes of this answer I will limit myself to the ashkenazi deli culture in greater New York.

There are two basic reasons for this phenomenon, one religious and one historical.

In Jewish dietary law, there are three basic categories of food: milchg (dairy products), fleishig (meat, excluding fish), and parve (foods that are neither meat nor dairy such as fruit, bread, eggs, and fish). The reason for this split is found in the Torah at Exodus 23:19, Exodus 34:26, and Deuteronomy 14:21, which instruct us לֹא-תְבַשֵּׁל גְּדִי, בַּחֲלֵב אִמּוֹ. - “thou shalt not boil a kid in its mother’s milk.” Kid in this context refers to a baby goat, human meat is not kosher regardless of cooking method. Talmudic authorities interpret this passage as a blanket ban on combining meat and dairy.

In practice, this means that observant Jews take a great deal of care to avoid accidental mixing, often keeping entirely different sets of plates and cutlery for milchg and fleishig to ensure that there is no cross-contamination. As you can imagine, this presents problems for restaurants, which might serve hundreds of meals a day and could easily result in an offensive mix-up.

The solution to this problem was to borrow a piece of wisdom from King Solomon and simply cut the baby (or in this case, restaurant) in half. It was simpler for early kosher restaurants to just pick either milchg or fleishig to serve exclusively rather than risk cross-contamination. Thus, the birth of the deli (which specialized in cured meats and cold cuts of various kinds) and its lesser-known fraternal twin, the appetizing store (which sold cheeses and preserved fish, along with the bagels and vegetable garnishes to serve them). Often, delis and appetizing stores would open up one right next to the other.

As time went on, New York Jews as a group became less observant and delis began to attract gentile customers, the need for a strict split between meat and dairy became less pressing. Appetizing stores started to die out, and many delis (particularly those outside areas with large orthodox populations) began to annex their adjoining appetizing stores and diversify into dairy and fish. A small number of appetizing stores survived, and the archetype has had a bit of a renaissance in recent years. Still, modern delis are more likely to be be “kosher style” than truly kosher.

The second main factor in the rise of the deli is the remarkably consistent pattern of development that immigrant cuisines go through in New York: from push-carts to store fronts in ethnic neighborhoods to sit-down restaurants with diverse clienteles. It’s been observed with Jewish, Italian, Greek, Chinese, and Russian groups, and now Arab and West Indian food are following the same trajectory. Today’s delis are the descendants of the wheeled carts and temporary street stands that sold pickles and sandwiches to Jewish laborers fresh of the boat from the old country. Some things don’t change, and we still love a good pastrami on rye.

Sources:

The Big Oyster by Mark Kurlansky

Brooklyn by Thomas J. Campanella

Metropolis by Ben Wilson

The Jewish Museum of New York