During combustion some of the gasses will get past the piston rings (called 'blow by') and go into the crankcase.
The purpose of the breather hose is to allow these gasses to vent and prevent pressure building up in the crankcase.
As there is negative pressure in the air box the gasses are sucked in with the fresh charge and burnt.
There are two spring clips fitted to either end, part #'s 90467-14015.
I don't know the diameter but if you measure the size of the hose an auto store will be able to supply something suitable.