From my observations of 18,000 puts:
Those lines mark the limits of an "acceptable" putt IMO. If you miss it it will be just crap, going only a fraction of the length.
When you miss within the lines, the putt will lose strength and miss to the same side. Some use this to aim on side slopes.
Obviously, it gets harder to stay in the "good" area when the outer lines have come closer. Then it's better to reduce the putting power to zero or just let it swing through without clicking, just to restart with full space. There's no (known) limit of repeating.