This is how I understand it.
Pretend you have a credit card. And this credit card has a limit, we'll say $1000. This credit card is pretty near maxed out and you don't really have any cash. You need to buy some stuff soon, and you know that between now and August 2nd you need to buy some things, and you have no choice but to buy them on the credit card. At that point the credit card will be completely maxed out.
This credit card is our debt ceiling. We will hit the limit of our borrowing limit on August 2nd.
Now let's continue further. We know we have some bills next month, and we also know that we have some cash coming in, but when we look at what we have coming in vs what we have to pay, we don't have enough to cover it. Let's just say we know we'll be short by $100. So now we know ahead of time that we'll be short, and we only have one real option: call the credit card company and ask them to raise our limit.
This is what the debt ceiling legislation is trying to do: raise our credit limit.
As you said, normally this happens all the time without issue. This time, some politicians decided to stand up and say: "Umm, long term this whole 'borrow more money' method may not work out." So they are holding off on raising the debt ceiling until we can better align our "bills" and our "income". There's two ways to do this: either you lower your bills or you raise your income. Either you pay less money out, or you bring more money in.
This is where the argument happens. Democrats (traditionally) would prefer to bring more money in, so they'd like to "raise taxes". Republicans (traditionally) would prefer to have lower bills, so they'd like to do "spending cuts".
So the argument now is "How can we find a compromise where everyone is happy?" We haven't (yet, hopefully) found that compromise.
If we don't find the compromise, and we don't raise the debt ceiling, then we'll have a bunch of bills due and not enough money to pay them. At this point we'll have to start prioritizing who gets the money we do have. Should it be seniors on Medicare? Should it be active duty military? Should it be people we owe interest to for a loan payment?
This is just like our credit card example if the credit card company doesn't raise our limit. Do we pay our rent? Do we pay our car payment? Do we pay back a guy we borrowed $50 from?
And the repercussions are this: whoever we DON'T pay, how does that negatively affect us? Will we be able to get more loans? Will people lose trust in us and a government? Etc. So the outcomes could be nothing or they could be disastrous. No one knows for sure.
There's more interesting discussion on the topic on the page.