The crankcase has to have a way to breathe, to allow the escape of the small amount of gas that leaks past the piston rings. The crankcase volume also varies as the piston moves up and down. There has to be a breather to accommodate all this, but for emissions control reasons, the leakage gases (known as "blow-by") are not allowed to be released into the atmosphere. The solution is to direct them into the intake pipe, so they get run through the combustion process along with the incoming clean air and fuel. The breather hose is just the way of directing surplus gas from the crankcase to the intake pipe.
How did you adjust the speed control lever? On Hondas it usually applies the choke when it is at the end of its movement, and you check this by setting the speed lever to the choke position, then pushing the control lever on top of the carburetor all the way to the right. If it was not already at its right hand stop, the choke is not being applied by the speed control when it is in the choke position, and you need to adjust the cables to make it work properly.
