“In a sense we’ve come to our nation’s capital to cash a check.”
What does it mean “to cash a check”?
A check is an old method of payment or money transfer that is not used much anymore. You write a check from your check book to someone for a certain amount of money. In order to get that money, the person goes to the bank to cash the check - i.e. they hand over the check and receive the amount of money that was written on the check.
Here is the wikipedia entry for check/cheque: Cheque - Wikipedia
In the UK we spell it “cheque”. 
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A check is an old method… Wow, I’m old.
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“to cash a check” can also be used metaphorically, meaning “to collect a debt owed”, “to ask someone to make good on a promise given earlier”
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At this URL http://waysandhow.com/how-to-cash-a-check-at-a-grocery-store you will see someone writing a check. To cash a check means to take a check that has been written to a bank and giving it to the bank in exchange for currency. The amount of currency received is determined by the amount of currency denoted on the check. Banks are institutions that cash checks, i.e. that accept written checks and give out currency in exchange for those checks. “we’ve come to our nation’s capital to cash a check.” This means that we have a piece of paper that says it is worth a certain amount of currency and instead of going to a bank we are going to the capital city of the country to exchange that piece of paper (that piece of paper=that check) for currency (currency=cash). I hope this helps…
Yes, it is an old method I am 34 years old and I have never owned a checkbook or written a check. I don’t think I have ever cashed a check except for some travellers checks over 10 years ago.
Do you still cash checks?
I have a checkbook, and I still use checks occasionally. In Brazil sometimes you can use checks for layaway payments. You give someone a few checks, and they hold on to them, cashing one every month. Some businesses prefer them over credit cards.