How can a massive star form a stellar-mass black hole at the end of its life?

Answer: Its core collapses beyond the support that can resist gravity.

Its core collapses beyond the support that can resist gravity. ✓ — When a massive star’s core loses support and the remnant is heavy enough, collapse can continue beyond what a neutron star can withstand. An event horizon then forms, preventing even light from escaping. This can happen with a supernova or by direct collapse without a bright explosion.

Its rapid rotation tears an empty hole in space. — Rotation does not tear an empty hole in space to make a black hole. A black hole is a region with mass concentrated enough to form an event horizon. Rotation affects its properties, but collapse supplies the mechanism in this case.

Its planets merge into a dense object after it cools. — Planets merging is not the process that forms a black hole from a dying massive star. The relevant mass is concentrated in the collapsing stellar core. If the remnant remains too light, it can instead be supported as a neutron star.