I made about $4500 last year as self-employed. This was my only income. My standard deduction is $6350. Why do I still owe $522 when I'm within the "tax-free income".

This would be true if your $4,504 of income was W-2 income and not self-employed income. On a W-2, your employer would pay for 1/2 of your social security taxes owed.

Since you are your own employer, you have to pay your 1/2 of self-employment taxes (SE) on your self-employed earnings of $400 or more. The tax is figured on schedule SE and you do get to deduct part of it (TurboTax calculates it for you).

https://ttlc.intuit.com/questions/1900698-what-is-the-self-employment-tax

https://ttlc.intuit.com/questions/1901205-where-do-i-find-schedule-se

The self employment tax is in addition to the regular income tax on it.  Which in your case the regular income tax IS zero.  Self employment tax is separate.

Self Employment tax (Scheduled SE) is generated if a person has $400 or more of net profit from self-employment on Schedule C.  You pay 15.3% SE tax on 92.35% of your Net Profit greater than $400.  The 15.3% self employed SE Tax is to pay both the employer part and employee part of Social Security and Medicare.  So you get social security credit for it when you retire.  You do get to take off the 50% ER portion of the SE tax as an adjustment on line 27 of the 1040.  The SE tax is already included in your tax due or reduced your refund.  It is on the 1040 line 57.  The SE tax is in addition to your regular income tax on the net profit. You will still owe the SE tax even if your taxable income is reduced to zero (by exemptions and deductions) and you don't owe any regular income tax on it.

You must have had some expenses to write off against it?  On $4,504 Net Profit I calculate the SE Tax to be about $636.

OP could have credits against SE tax, for example EIC.