Well i always said the ATmega 0 series was being done in ARM style but was laughed at.
But no this makes no sense. A register is a memory location. It tells the peripheral how to behave. If I write a 1 to the set the output in one register, and then write a 1 to another register to clear it there is a conflict.
Now the single 0 or 1 state of one pin is determined by 4 combinations from 2 bits of which 2 never occur. There has to be something, when you set both registers to "1" how does it know what to do?
Although ataradov answered this partially, this is a crucial difference to understand.
In traditional registers, as you understand, you have a set of bits in hardware that you can read/write, and then used by the peripheral to do stuff with. These registers are then traditionally implemented as D-flipflops, i.e. the peripheral address is matched from the bus, and then that value written is put into the D flipflop.
However in this case you have SET and CLR registers for port operations. You could view these registers as write-only. Picture that they are implemented as SR flipflops instead of D-flipflops. (or technically JK flipflop because we're dealing with synchronous, clocked, systems).
If you write to the SET register, those bits are wired to the set input of the flipflop. And the CLR it's wired to the reset flipflop.
It doesn't require this implementation, but I hope it helps in understanding.
This SET/CLR structure avoids the CPU having to perform a read-modify-write operation (i.e. an bitwise OR or AND operation with a particular bitmask), as it could be handled by peripheral internally. In particular, you avoid the "read" operation from the bus and only a write operation remains.
This is also useful because bitwise I/O operations are now atomic, i.e. you don't have to guard them when you need to deal with interrupts.
For ARM chips, cortex m3 and "up" features bitbanding. Here the whole peripheral and memory space is mapped into a separate piece memory region, where you can address each bit as an individual word, and read, set or clr them.
Also, this is not a particular feature of ARM. PIC32 also features these kind of registers, because on 32-bit CPUs as said, all I/O is done via RMW operations and thus painful.