Why would the check engine light go out without making a repair?

Basically it means that you have experienced an intermittent emission control system failure. It failed to work properly for some time, then it began working again. Explanations range from a temporary issue caused by wet or dirty wiring harness plugs that may never give you a problem again, to a component that is slowly failing and will definitely have to be replaced, eventually.

For an explanation of how this can happen, see below:

The check engine light illuminates when an emission control device has failed a computerized self test. Such tests vary by year and by car, but they are basic operational tests that occur all the time your vehicle is in operation.

During vehicle operation, the ECM (electronic control module) reads a series of sensors (temperature, pressure, flow, position, etc) and controls a series of effectors (valves, switches, solenoids, etc). If the ECM turns on a solenoid to effect a change in exhaust gas recirculation, it will watch the exhaust gas recirculation flow sensor to see if the change actually created more or less flow. If it does not, it may reset the solenoid and try it several more times. But eventually, if the effector (switch, solenoid) does not cause a change in the reading of a sensor, the ECM will mark the circuit as defective, set a service code(s), and illuminate the check engine light.

Additionally, the ECM will occasionally retest any circuit marked defective. If that circuit works properly during the restes, it will mark the circuit as good and turn off the check engine light. These retests occur when you first start your engine, but may also happen while you are driving. The newer the vehicle, the more sophisticated and smarter the ECM is.