The Fed issues as much dollars as needed by writing the new trillions on its balance sheet. These new trillions are channeled usually via government bond buying or through the banking system.
The US government designs its economic policies in a way that makes its trade balance of payments always negative (by large values). Therefore, trillions keep flowing overseas (to pay China, EU, Japan etc for the imported goods and services).
Moreover, the US coerces foreign governments to trade between each others in USD; especially for oil and gas. In this way, the ‘excess’ dollars keep circulating worldwide.
Furthermore, the US maintains large military (800 bases) and diplomatic presence. In addition to its main mission, this presence helps also diluting trillions all over the world.
As long as this system continues working, hyperinflation is very unlikely to happen.
But if the US starts exporting more goods than it imports, or foreign governments stop using the USD to trade between each other, big collapse will certainly happen.