Diagnosis
The clamp spring or the hinge mechanism is dislodged or broken. This is a physical failure.
How to Fix It
Unplug the unit. Examine the silver clamp arm. Where it connects to the black housing, there is a small metal pin. Ensure it is seated in its hole. If the pin is out, you can try to press it back in with pliers. If the clamp feels springy but won't stay shut, the internal torsion spring has likely slipped off its post.
If That Didn't Work
The clamp assembly is a single part. It is not designed for user repair. You need a replacement curling barrel. The clamp is built into the barrel unit.
Heads Up
Do not use the iron with a malfunctioning clamp. Hair will not feed correctly and will wrap around the hot barrel, causing damage and potential burns.