TWRP is an open source, community project. TWRP development is done by roughly 4 people at this point. We also have a large support community with many people who are willing to answer questions and help people with their devices either through our Zulip channel or on forums like xda-developers.
Team Win was originally formed to work on porting WiMAX to CM7 for the HTC EVO 4G. After our work on the EVO 4G we wanted to work on a project that would work on more devices than just the EVO 4G and we settled on working on a recovery. Today TWRP is the leading custom recovery for Android phones.
A custom recovery is used for installing custom software on your device. This custom software can include smaller modifications like rooting your device or even replacing the firmware of the device with a completely custom "ROM" like OmniROM
You can find the source code for TWRP at github.com/teamwin/android_bootable_recovery/