They are mixing two different things here.
The strong force does not work between protons and neutrons, it works between quarks. As a side-effect of the way it works, it also constantly creates new particles, mesons. This particle-creation process is conservative in that if you consider all of the particles that are created, their momentum, charge, spin and so on all add up to zero.
It's all those mesons interacting with each other and the quarks that give rise to a second force, the nuclear force. It is the nuclear force that "acts on protons and neutrons to keep them bound to each other inside nuclei", not the strong force. Of course, one is the result of the other, so half full sort of thing... and that's why you see it called the strong nuclear force, or the strong interaction or all sorts of other names just to confuse things.
The distance that the nuclear force operates over is simply a function of the mass of the mesons and the uncertainty principle; all virtual particles with mass have a maximum lifetime, and if you simply see how far these mesons can go in that time, presto, you get a distance.
UPDATE: I realized I missed the closure.
The weak force is also mediated by massive particles, the W's and Z's. Like the mesons in the nuclear force, they are thus subject to the same range limitations due to the uncertainty principle. However, the mass of a simple meson like a pion is about 100 MeV, while the Z is a whopping 90 GeV. That's heavier than an entire iron nucleus! Now it might sound odd that such a heavy object can be created ex nihlo inside something like a helium nucleus, which is way lighter, but that's the whole idea of the uncertainty principle, for a very short time this is allowed, and that's why it's range is so short and the reaction is so rare in comparison.