Whats this do? And where does it go?

Got a new to me 2001 Buell Blast. Bike has decently low miles (6200) but was sitting idle for a couple of years. Previous owner said he tried to get it running, but was unsuccessful. Bike had spark and turned over but seemed like it wasn't getting gas. I charged the battery and took the carb apart and cleaned it (all jets except the one under the float, couldn't figure out how to get it off, so any suggestions on this would be helpful.) Doing so I saw a hose inside the air box that looks like its supposed to attach somewhere but I don't see anywhere it can. It also has a T fitting that has one spot that nothing is attached. Where is this supposed to attach to? And why is the T fitting only connected on 2 sides? Makes no sense to me. I did get the bike to start and idle but it stalls when I apply the throttle. I could move using very little throttle as long as it I didn't exceed 10 mph. I'm hoping it's an easy fix and that the hose is supposed to attach somewhere that I couldn't figure out. Any help on any of this is appreciated. Oh and if someone can tell me how to post pictures I can provide some of what I am referencing.

Find a sevice manual, 'sounds like a dis-connected vacuum hose. Look for a bare barb somewhere....

The open line in the airbox is the breather. It's supposed to be open, venting the crankcase fumes toward the carb to be re-injested by the engine.
Your carb isn't giving the engine enough fuel because you didn't clean the WHOLE THING.
Theres no "Jet" to clean under the float (the float comes out by pushing the little pivot pin out, careful not to bend the tabs that dictate the float level.
Both jets (the pilot jet and the main jet need to be removed and cleaned at a minimum. Welding tip cleaners (Harbor freight) are great for that. They are both accessible from the float bowl.
Do yourself a favor and just do the whole thing. Get a carb kit ($9), instructions, and a tub of Berrymans. You will thank yourself for doing it right. Once.