It sounds like you did not read his policy. It was very well thought out and the source of the UBI was going to come from a combination of a strategic VAT and also by removing idiotic subsidies like the ones we give the oil industry, etc... So no, it does not come out of thin air. Also, the $4 Trillion number does not take into account the saving on welfare and other programs that would be significantly lessened - the number is more like $2.5 T.
We currently give the subsidies to the rich who may or may not spend it back into the economy. You give that cash to the masses and nearly all of them will spend it immediately - putting it back into the economy - which would stimulate the economy.
UBI does not mean everything will be more expensive, it means that consumers have more power and more choice. instead of renting, people can pool their UBIs and buy a home, or they can move to a different area with more options. It is just an economy here income does not start at zero.