The words "accept responsibility for paying it" describe a situation like the following: I get a check from Mr. Cheatum for $10 because I mowed his lawn. The "check" is a typical printed 6x3 inch (15x7 cm) piece of paper with Mr. Cheatum's name and address in the upper left corner, a check number in the upper right corner, a "memo line" and bank information in the lower left corner and a signature line in the lower right corner.
In the middle of the piece of paper there are, on the left side, the words "Pay to the order of" followed by a long underscore that extends nearly to the other side of the piece of paper. After the underscore there is usually a dollar sign ($) and another underscore.
On the next line there is a long underscore from the left side that ends in the word "Dollars".
These two central lines are filled in by Mr. Cheatum. After the words "Pay to the order of" he writes my first and last name. Then he writes "10.00" after the dollar sign.
On the next line he spells out "Ten and no/100".
I take that piece of paper to my bank, sign ("endorse") it on the back (usually adding, for security, "For deposit only into XYZ bank account number 123456 above my signature). The bank teller takes that check and, pending the check's clearance, credits my account for ten dollars. The operative words in that last sentence are "pending the check's clearance."
Sadly, Mr. Cheatum doesn't have enough money in his account (or doesn't even have that account anymore) to cover this check.
Now my bank comes back to me with a form letter saying something like, "The enclosed check from Mr. Cheatum was returned for insufficient funds. XYZ's policy on returned checks is to charge a $35 fee for each returned item".
I now am out the $10 that Mr. Cheatum didn't actually pay me, and the $35 that the bank charged me for the trouble of putting the check through the system and not being able to credit my account.
That's what "accept responsibility for paying it" means. The bank took the check, but holds me responsible for its value. By signing the back of the check I have indicated to the bank that they can credit my account $10 because Mr. Cheatum's signature is now "backed by the full faith and credit of ... me"--I "endorsed" Mr. Cheatum.
This whole situation is relevant almost exclusively, I believe, to the United States, where checks (sic) are still surprisingly common.