Can someone help me understand why I won't receive the full child tax credit amount for my 3 children?

The CTC is limited and can only reduce your tax to zero.  In your case you would have to owe $6,000 in taxes.

However, You may qualify for the Additional Child Tax Credit.  This is not limited to the amount you owe in taxes.

• ACTC, is worth up to $1,700 for each qualifying child.
• ACTC is not allowed if you or your spouse (if filing a joint return) file a Form 2555 (Taxpayers that can exclude income earned in foreign countries.)

You can check your tax return and verify if you did get the ACTC 

From the left sidebar

1. Select Tax tools
2. Hint: You may have to minimize the Federal and State selections
3. Click Tools
4. A Menu will pop up
5. Select View tax summary
6. On the left panel click Preview my 1040

Some Key numbers:

• 11 Adjusted Gross Income
• 12 Deductions
• 15 Taxable income
• 16 Tax
• 19 Child Tax Credit
• 28 Additional Child Tax Credit
• 33 Withholdings
• 34 Refund
• 34 Underpayment

See the TurboTax article What is the Additional Child Tax Credit?