For one, it isn't always institution A that requires this. It could be that they may come to the FED for help more often but if that occurs then the FED will either deny them or ask them to take less risk. This happened with Bear Stearns back in 2008 where the FED denied them access to repo facility if I remember correctly.
To understand it a bit better, look at the repo facility as a bridge loan for individuals. In this scenario you are a bank and your friend is the FED. Say you make $2,000/month and you have $2,000/month in expenses. For whatever reason you are sick on the day you're supposed to get paid and do not pick up that check. But your cable payment of $100 is due that night, so you go to your friend and offer him your brand new macbook as a collateral which is worth at least $800 on Ebay. You agree to pay him $110($10 in fees) back on Monday when you get to work and pick up(and cash) your check. He agrees to return that macbook to you if you pay him back $110. You do this several months in a row. But now your expenses are $2080($80 in racked up fees for using this bridge loan facility) when your income is $2000. Your friend realizes that you cannot sustain this forever so that's when your friend asks you to cut down on your smoking/drinking/entertainment expenses so you can recalibrate your income and expense. You do it and everyone walks away happy...in principle.
For everyday Joe, if this gets out of control the biggest concern here is that it reduces liquidity in the market, which means less capital available or capital available but with a high downpayment to open or expand businesses. The inability of businesses to get new cheap debt causes a slow down which means fewer jobs which means fewer opportunities which will eventually cause a recession or worse.