How can I unscrew the faucet nuts in the tight space behind my sink basin?
This is exactly the purpose of a basin wrench 

 

Here's a picture from familyhandyman.com that shows a basin wrench in action:  



Note that as mentioned in another answer, you usually don't need to remove the faucet to replace the washers;  but if you do need to remove the faucet for whatever reason - this is the tool.