How did the construction of the colosseum demonstrate the flavian dynasty's commitment to the well-being of the people?

It wasn't so much for the well being of the people. Despite what many people think, in the city of Rome itself, the mob ruled. By building the Colosseum, the Flavian Dynasty gained a way to make the people happy, by providing them with games. The bonus was that some jobs opened up while they were building it, but the majority of the construction was handled by slaves and freemen.