It's the coolant reservoir. In case the pressure exceeds the rating of the cap, the hose is meant to carry the overflow. Yes, it's just open on one end. The reservoir may have been overfilled at one point, or coolant may have gotten hot enough that it expanded and overflowed. I wouldn't worry too much as long as the coolant level is ok and you don't keep having leakage.
If you have persistent leakage there and are losing coolant, inspect the cap for proper fit (when the system is cold!) and make sure the seating area is clean. Pressure caps can develop a leak. Easy enough to replace with a new one if it's not holding pressure. I guess it's possible to have overheating problems but I've never heard of it in this vehicle.