The problem that turntables solve is that the inside of a microwave oven is a resonant cavity in which constructive and destructive interference of the microwaves results in regions of high and low fields, causing hot and cold spots in the food. The simple solution is to move the food during cooking. This can be done by hand, by stopping the microwave, opening it, moving the food to a different position, and then cooking/heating it further. The turntable does this simply and automatically.
The turntable appeared before the microwave oven became common in the home, introduced by Sony in the mid-1960s. The first popular microwave oven is usually said to be the Amana Radarange, introduced in 1967 (produced by Raytheon, developer of the first microwave ovens, who had bought Amana). The Amana Radarange had no turntable. Instead, a different solution to hot and cold spots was used: the antenna delivering the microwaves into the oven cavity was moved by a motor, shifting the interference pattern and having a similar effect to moving the food. Some ovens used a third option: a fan-like paddle rotated, shifting the interference pattern by changing the geometry of the cavity.
Despite these three different methods, many microwave ovens were made without any of them, and required manual moving of the food to avoid hot and cold spots. As microwave ovens became more popular in the 1970s, after-market turntables that could be used in microwaves were introduced:
During the 1980s, turntables became more common, eventually becoming almost universal in home microwaves. More recently, turntable-free microwaves have become popular (usually using a stirrer, the last method of shifting the interference pattern noted above). Turntable-free microwaves were always popular in commercial settings, since they are easier to clean. The stirrer also produces more even heating than a turntable.