I've heard this in many quantum mechanics talks and lectures, nevertheless I don't seem to grasp the idea behind it.
There is no “idea” behind it. Just general fluff talk, like saying we wouldn’t have the lightbulb without Ohm’s Law (Fact: Lightbulbs were in existence before Ohm formalised his law, and the first practical Swan-Edison bulbs had one feature—high resistance—derived from Ohm’s law, among many other innovations. Ohmic resistance is not essential for a lightbulb at all).
What I mean is, at which point is that our modern understanding of quantum mechanics led to a technological development so fundamental for today's computers that we could not have got it working other way?
You’re mixing the existence of quantum mechanics with understanding of it. In Physics, theory usually follows observation (except for a few dramatic cases). Today’s computers depend on semiconductor-transistor action, which is explained by QM theory. QM theory enabled the refinement of transistors (eg: predicting what doping materials would produce what effects on which substrates). This, while good, is not fundamental to computing or computers at all. Whether we could have had it working any other way? Definitely yes. Whether it is possible to compete with today’s computers with non-semiconductor tech? That’s a hypothetical question! Could the Allies have lost WW II? Yes, but they didn’t. So it is with semiconductors—they were (and are) the best available mechanism for cheap, ubiquitous computing.
Why is it not enough with Maxwell, Bohr, Lorentz, (Liénard)?
It’s enough with Newton, Coulomb/Gauss, Faraday, and van der Waals. It only depends on what your definition of “modern” is. For example, imagine an advanded Babbage machine built with nanoparticles, featuring molecular gears and cogs. Now imagine said machine with electrical linkages (using dynamos and capacitors) to mimic a CPU pinout. Can this machine replace a Core i5 and run Facebook? Absolutely. Do we have such machines? No.