You seem to be very confused about taxes. The withholdings from your paychecks are just the result of a reasonably educated guess of what your total tax liability will be at the end of the year. Your actual taxes aren't actually calculated until the end of the year when you gather up all of your w2s, 1099s and other documentation and file.
US Federal Taxes brackets indicate marginal increases, meaning (for single folks) you pay 10% on income up to $9,275, and 15% on income from 9,276 to $37,650 and so on. The $9,275 figure is based on taxable income, meaning after you calculate all your income and all your deductible expenses. There is also something called a Standard Deduction. Unless you have itemized deductions in excess of the standard deduction you'll just take the standard deduction.
Say person A makes $10,000 in a year. "A" uses the standard deduction of $6,300, which makes his taxable income just $3,700. He will owe $370 in federal taxes. This tax liability represents just 3.7% against gross income of $10,000.
Say person B makes $20,000 and also uses the $6,300 standard deduction making her taxable income $13,700. She will pay $1,591.25. This is calculated as 10% on the first $9,275 ($927.50) and 15% on the remaining $4,425 ($663.75). This tax liability represents 7.9% against gross income of $20,000.
For two times more income ($10,000 vs $20,000) you pay approximately 4.3 times more in taxes ($370 vs $1,591).
In your question you're showing July monthly hours of 40 and August monthly hours of 173.33. At $15 per hours these will represent approximate annual incomes of $7,200 and $31,140 respectively (40 hours times 12 months times $15 per hour). These two annual incomes will have drastically different tax liabilities.
Your employer is taking your income in a given period then assuming you will continue to earn at this same rate for the entire year. From that assumption it's calculating a tax withholding. But your employer doesn't know if you have another job or do some contract work on the side or own a home or have kids or any number of other things that could change your tax situation. This assumption and corresponding withholding is very rarely exactly correct. At the end of the year you will file your taxes and either owe a bit more or receive a refund.