As others have explained this pin allows the 68000 to select an individual byte to read/write rather than a 16 bit word. However, they haven't explained why it was designed this way.
The 68k was designed for a 16 bit bus, with 16 bit RAM and ROM and peripherals. Byte addressing is not used with such a system, the smallest addressing unit is a 16 bit word. If the 68k wants to read/write a byte it handles it internally, e.g. reading a word and discarding 8 bits.
This technique is problematic for 8 bit peripherals. A read operation can have consequences as some parts alter their state on a read, e.g. a FIFO, so having to read an unrelated register or do a read-modify-write cycle would make the 68000 incompatible with them. The solution was the byte select pin, which acts as an additional address pin for 8 bit devices on the bus.