I'm a self employed freelancer who is paying quarterly taxes. The IRS shows that I can direct pay online, on their website. Do I have to send in form 1040ES, or do I just pay the amount I calculate? I'm confused because I have a deduction for a home office, but how will they know I used this deduction if I just pay online?
The actual form 1040ES that you might file is at the bottom of page 9 of this PDF (and repeated for each quarter). On the form it says:
File only if you are making a payment of estimated tax by check or money order.
In general, you don't have to file any form to make the estimated payments. You just have to make the payments. After the tax year ends, you will file your usual tax return (e.g., form 1040). There you will indicate how much you have paid in estimated taxes, along with all the usual deductions and so forth, and based on that you will determine if you owe any additional taxes or will get a refund.
how will they know I used this deduction if I just pay online?
They don't, because you haven't used the deduction yet. You'll use the deduction when you actually file your taxes next year. Your estimated payments (based on the various worksheets included with form 1040ES) may incorporate your intention to use that deduction, but the estimated payments are just an estimate. The actual determination of tax due is still done at the usual time, after the tax year is over.