Hi Amy. If your total, taxable income is $15,000 and, now, you set aside $225 per month tax-deferred, then your total, taxable income goes down by $225 x 12 = $2,700 per year. So, your taxable income becomes $15,000 - $2,700 = $12,300.
According to the tax table you had copy-pasted, one pays 10% of the first $9,950 of income in taxes, then 12% of the next $40,525 - $9,950 = $30,575 of income.
So, if your total, taxable income is/was $15,000, then the taxes on that would be $995 + $606 = $1,601. That means annual take-home is/was $15,000 - $1,601 = $13,399, which comes out to $1,116.58 per month.
Now, if your total, taxable income has become $12,300, then the taxes on that would be $995 + $282 = $1,277. That means annual take-home has become $12,300 - $1,277 = $11,023, which comes out to $918.58 per month.
Notice how the $2,700 drop in taxable income -- because your retirement plan contributions are coming out of the same, annual $15,000 salary -- notice how the shifting of income away from taxable income to a retirement plan is also saving you $225 - ($1,116.58 - $918.58) = $27 per month in taxes.