Disclaimer: not a physicist, and I never went beyond the equivalent to a BA in physics in my formal education (after that I "fell" into comp sci, which funnily enough I find was a great pepper for wrapping my head around quantum mechanics).

So space and time per se might be continuous, but the energy levels of the various fields that inhabit spacetime are not.

And since, to the best of our current understanding, everything "inside" the universe is made up of those different fields, including our eyes and any instrument we might use to measure, there is a limit below which we just can't "see" more detail - be it in terms of size, mass, energy, spin, electrical potential, etc.

This limit varies depending on the physical quantity you are considering, and are collectively called Planck units.

Note that this is a hand wavy explanation I'm giving that attempts to give you a feeling for what the implications of quantum mechanics are like. The wikipédia article I linked in the previous paragraph gives a more precise definition, notably that the Planck "scale" for a physical quantity (mass, length, charge, etc) is the scale at which you cannot reasonably ignore the effects of quantum gravity. Sadly (for the purpose of providing you with a good explanation) we still don't know exactly how to take quantum gravity into account. So the Planck scale is effectively the "minimum size limit" beyond which you kinda have to throw your existing understanding of physics out of the window.

This is why I began this comment with "space and time might be continuous per se"; we just don't conclusively know yet what "really" goes on as you keep on considering smaller and smaller subdivisions.