I'm no physicist, but the layman explanation I've heard is that the concepts of Planck length and Planck time do not imply that spacetime is discretized into little "voxels"; instead these quantities are just limits on the uncertainty of any possible measurement.

As I understand it there are two contributing facts:

a) the Heisenberg uncertainty principle states that there's a tradeoff between certainty in position vs momentum, so if you're more certain in a particle's position you're less certain of its momentum. (For photons, momentum is proportional to frequency, i.e. wavelength, and frequency is proportional to energy.)

b) By mass-energy equivalence, anything with energy has mass, therefore higher frequency photons are more "massive". A single photon of sufficiently high frequency would form a black hole.

Putting those two together, to measure a distances accurately, you need higher and higher frequency photons with shorter and shorter wavelengths. For example, radar creates blurry images at ~5cm wavelengths, while ordinary photographs can be razor sharp at ~500nm. The Planck length is just the wavelength at which the photon would have so much energy that it would collapse into a black hole and break our current mathematical models. That's why it's nonsensical--with current models--to talk about lengths smaller than a Planck length, but it doesn't mean that space itself is quantized. Similar argument for time.

(Also, the same logic applies to other particles like electrons, protons, and even up to macroscopic items like baseballs; everything has a wavelength...)

> I've always thought that Planck's constant defines the discreteness of time

Most certainly not! Amusingly, this is an incredibly common assumption that undegrad physics students I teach make, probably because of some pop-sci exposure.

All we know is that the Planck constant defines a scale where we do not know what happens. This is a scale at which we can see that the math behind our current theory breaks, but we have absolutely no reason to expect that the way to fix that math is to use some form of discretization related to that constant.

The "quantum" in quantum mechanics really should not be taken that literally. Photons as described by quantum mechanics for instance do not have a discrete spectrum (nothing literally "quantum" there).

There are fascinating conjectures on why maybe the Plank scale should be discrete, but they are way more subtle than "quantum mechanics is discrete" (because it is not always discrete).

> is as fundamentally important to the fabric of the universe as it's much more well known "brother", the speed of light in a vacuum c.

It is not. It would violate special relativity. No such violation has ever been observed.