Let's say for example, oh I don't know, you're riding your bike on rural roads 20 miles north of Nashville while your inlaws are back at the rental house downtown on Thanksgiving day, and your back tube blows out. Let's just say that happens.
And then (because you should always figure out what made the flat happen) you see that the sidewall has been compromised and that's what caused the tube to blow. And if you simply put a new tube in, it'll blow again shortly - like it'll blow way before you get back. It looks like this:
Chances are you don't want to pay the cab fare home, or call your wife or inlaws who are preparing Thanksgiving dinner to come pick you up.
You have to fix it. And here's a way to do it when you're in a pinch.
1. Cut the old tube in half.
2. Cut off a piece 2 or 3 inches long.
3. Cut a slit in that so it can cover the part of the tire that has the compromised sidewall.
4. Carefully change and inflate the new tube so that the old tube "patch" stays well in place.
5. Ride carefully, take turns slowly without leaning over, avoid holes in the road.
I was able to get home with this patch. The best thing to do is keep 3 inches or so of old tire with you on these sorts of rides. If the sidewall had larger damage, this sort of patch wouldn't have worked. But it's a good little trick to know (especially when you ride tires to death).