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TestFlight is an add-on for Kerbal Space Program (a mod) that simulates the effect of increasing the reliability of your space hardware through flight testing.
Wikipedia defines Flight Testing as:
Flight testing is a branch of aeronautical engineering that develops and gathers data during flight of an aircraft, or atmospheric testing of launch vehicles and reusable spacecraft, and then analyzes the data to evaluate the aerodynamic flight characteristics of the vehicle in order to validate the design, including safety aspects.
The flight test phase accomplishes two major tasks: 1) finding and fixing any aircraft design problems and then 2) verifying and documenting the vehicle capabilities for government certification or customer acceptance. The flight test phase can range from the test of a single new system for an existing aircraft to the complete development and certification of a new aircraft, launch vehicle, or reusable spacecraft. Therefore, the duration of a flight test program can vary from a few weeks to many years.
While I didn't grab that quote from Wikipedia until after writing TestFlight, I find that it does an incredibly good job of summing things up, especially the parts I highlighted in bold.
In a nutshell TestFlight is a persistent, parts based research and reliability program. Your parts start out with an inherent failure rate that might be quite high on some parts, and quite low on others. As you fly those parts, you gain flight data. If the parts fail, and they will, then one of many failure modes can occur, some that can be repaired and some that can't. Either way you will accumulate data from failures and repairs as well. All this data is transmitted back to your spaceport, and the next time you fly the same hardware it will have a lower failure rate based on how much flight data you collected in the previous flight. This happens on a per part basis, along with RealFuels integration so that if you use RealFuels to change the engine to a different config derivative, that derivative will be tracked as well.
If I could sum it up in a picture it would be this one which tracked flights from the early United States space program.
All those red marks on the left side are failed launches. The rest are successes. This is the gameplay TestFlight will give you. Early launches will fail, but they will get better as you launch the same hardware multiple times.
For a lightweight alternate see TestLite