GDP is the sum total of every dollar spent on every good and service in a country. I doubt the US government can keep track of all such transactions, so what do they do to estimate GDP?
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The basic GDP formula is GDP = Consumer spending + Investments + Government spending + Net exports.
The government obviously knows its own budget, so that part is easy.
Consumers spend stuff in shops, restaurants and other places which probably pay taxes. The government knows how much all these places earn.
Companies invest money. The total amount can be obtained by asking banks and other investment funds how much they’re giving or by looking at companies budgets and tax filings.
As for exports, you have to declare the value of your stuff when it crosses borders.
There are other ways, like surveys, but a decent country with functioning tax systems and border control can get a pretty good idea of their own GDP.
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