What is the significance of Planck units?

All Planck units are constructed from $c$, $hbar$, and $G$. The speed of light $c$ is the defining constant of Special Relativity. Planck’s constant is the defining constant of quantum mechanics. Newton’s gravitational constant $G$ is the defining constant of gravity.

Thus any theory which brings together Special Relativity, quantum mechanics, and gravity is going to presumably involve Planck units. The constants going in to such a theory are going to come out in Planck combinations when you calculate something, based on nothing deeper than dimensional analysis.

(But perhaps there will be surprises such as huge dimensionless factors throwing all the Planck scales dramatically off! As a made-up example, suppose spacetime quanta, if such things exist, are not on the Planck scale but a factor of $e^{pi/alpha}$ smaller, where $alpha$ is a small fundamental unified coupling constant similar to the fine-structure constant of electromagnetism. Can you see what a huge difference that would make? And what it would mean for trying to test the theory experimentally?)

No theory unifying Special Relativity, quantum mechanics, and gravity currently has wide acceptance, so the actual physical significance of Planck units is unclear. People talk about things like “spacetime foam” or “spacetime quanta” at (around) the Planck length and time scales, string-theory particles with masses at (around) the Planck mass or higher, possible unification of all forces at (around) the Planck energy, black holes containing Planck stars having (around) the Planck density, and the like. But in the absence of an accepted theory of relativistic quantum gravity, all these things are speculative. They can be exciting, fascinating, and beautiful ideas, but alas they are speculative.

They have been speculative for all of my life, and I expect that they will remain speculative for all of yours. (I am old enough to remember the “first superstring revolution” in 1984. Here we are 36 years later. Lots of interesting progress, but no accepted theory of relativistic quantum gravity, using strings or anything else.) But I would love to be wrong about this!