With apologies for reviving an old thread, it looked like gary350 had almost worked out the correct answer before he was given well intended, logical, but incorrect explanations that he apparently accepted at face value.
The reason the tip of the soldering gun heats up rather than the rest of the wire isn't because the copper wire is cracked or stressed at the tight bend. Its because the magnetic field created by electrons traveling in a conductor causes them to repel each other, pushing the current to travel at the surface, this is called the skin effect. The skin effect is always present, but its more pronounced for AC than DC, and especially at high frequencies or very high current. The tight bend at the tip puts currents flowing in opposite direction in close proximity causing an intense skin effect there, the traveling electrons use just the surface of the copper, only a few atoms deep. The internal bulk of the copper isn't carrying current, as if it were hollow. This is why that area presents a high resistance to the current and heats up far more than the identical wire nearby.
As the tip ages it may (will) also develop stress cracks as the differential heating near the tip causes mechanical stress. Even a small surface defect like a teeny corner crack will present disproportionately higher resistance causing locally more intense differential heating, promoting rapid crack growth. This is the most common operating failure, while being mashed in the toolbox appears to be common non-operating failure.
The skin effect is a well worn topic for power transmission systems and RF equipment among others. Engineers are usually more interested in calculating it to avoid it than to exploit it.