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The brackets are pretty simple. It's percentages and subtractions. Think "buckets" that spill over in the next when they're full, and each "bucket" has a larger percentage that's taken as taxes. Keep the numbers small so its easier. Imagine that there are three brackets. 0-100$ pays 10% tax. 101-200$ pays 20%. 200$ and more pays 30%.
Someone who wins 150$ pays 10% on the money they made from 0 to 100$, and 20% on the 101st dollar until their last, so they'll win 150-10-10=130$ after tax. They didn't win more than 200$, so no money gets taxed at the third bracket's rate.
Say that person wins 250$ next year. Their first 100$ will result in the exact same 10$ in taxes. Their 100th to 199th dollars will be in the second 20% bracket. Their remaining 50$ falls in the last bracket, so gets taxed 30%. They will therefore this year make 250-10-20-15=205$ after tax.
Said person gets a big promotion and is now making 1000$ the third year. Their first 100$ gets the same 10$ tax, same for their second 200$ with the same 20$ tax. They have 800$ left in the last bracket, which at 30% means 240$. So they'd be winning 1000-10-20-240=730$ that year.