Like many things in the United States tax code, the foreign tax credit computation is not a simple straight forward calculation that provides a credit for the amount of tax paid to the foreign country.
The credit is designed to cover the "allocated amount of U.S. tax imposed on the foreign income", which means that the calculation of the credit determines how much of your overall U.S. tax bill for the year is attributable to your foreign tax. If you have income that is a mixture of U.S. sourced and foreign, then it is often the case that your credit will be for less than the full dollar amount paid in taxes to the other country.
For example, if you paid $2,000 to France on $30,000 of French income but you have $90,00 worth of total income on your tax return, then your tax credit is likely to be around 1/3 of the tax that you paid because the formula will have determined that the French income is attributable to approximately 1/3 of your total United States tax bill. There are a number of other deductions and items that go into the credit, so your numbers may not be that round, but that's in very basic terms how it works.
If you want to look at the full methodology for the calculation (or you just can't sleep tonight) you can look at the instructions for Form 1116 to follow how the credit is determined.
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