Looks like you need to replace the washer at the top of the valve stem (this pic is of an anti-freeze faucet):
I know this is not the same type of valve as yours, but the idea is the same:
Remove the handle (using the screw in the middle), then use a wrench to remove the packing nut. The packing washer is what you'd need to replace. Most any hardware store/home center should have the kind you need.
It works fine with no hose connected b/c there is no back pressure -- water will take the path of least resistance, so it's going right out the spigot. A hose increases resistance, and will force some water to seek another path, in this case out around the packing washer.
Unless I've misunderstood where the water is coming out..