Q: The RPMs are extremely high when I try to shift gears.

For about 2 weeks now, the RPMs are very high when I shift gears. It sounds like I'm going way faster than what I actually am. This same thing happens when I drive uphill. I checked my transmission fluid just to rule something out, and the levels are fine. I also tried putting my car in a higher gear with the parking break on, and the engine doesn't stall when I let off the clutch. I don't hear any grinding when I shift gears. I'm not very car savvy, but I had my mom look under the hood while I shifted gears (the engine was off and the parking break was on) and found the box where the gear shift must happen. There is a mesh-type tube with a white plastic top that looks like it's supposed to screw in somewhere that is just sticking out. I believe the tube is coming from the box where the gears shift. I have pictures as well of the tube. Could this be my problem? Thank you so much!

My car has 138000 miles.
My car has a manual transmission.