What is the mechanism that causes two conductors to "stick together" when a current is passed?
What is the mechanism behind this?


Microwelding.

The point of contact between the two leads has a significantly higher resistance than the wires themselves, or their connection to the power supply. Passing a current through this circuit will cause heat to be generated at that point of contact, melting some of the metal around that point and causing the leads to stick to each other.

If you're using an inexpensive bench power supply, I wouldn't be surprised at all if its current limiting were not responding fast enough to a sudden short like this. The actual current that flows when the leads first make contact may be significantly higher than the set limit -- depending on the exact architecture of your PSU, it's possible that its output smoothing capacitors are being discharged directly into your test leads, bypassing current limiting entirely.