This game is an unlicensed version of Portal 2.
Thinking with Time Machine is a Portal 2 modification featuring a new gameplay element known as the "Time Machine" which allows the player to record and play back their actions in the game world.
The story continues from the end of Portal 2 where Chell successfully escapes the facility. However, a crow accidentally flies into the "sky", breaking the illusion of a wide open field created by giant screens. After Chell reawakens, she is revealed to be in the underground portion of the facility, where an elevator leads to new test chambers.
After completing a tutorial, Chell runs across an unsolvable test, until you press a button and a record player falls out, followed by a Borealis hologram, which cracks a hole in the wall. Chell discovers the two co-op bots, saying that they shut down GLaDOS. They say they will keep tabs on Chell, and the new "Time Machine". GLaDOS tells Chell to go back in time, and the chamber opens.
She is then tasked with completing 9 highly challenging test chambers. After their completion, Chell reaches a white spherical room. It transports her back in time to Portal 2's ending, at which point GLaDOS tells her that she is 'free'.
Source: The Portal Wiki, "Thinking with Time Machine", available under the CC-BY License.