What you need to realize is that coaxial cable is a waveguide, and the wave is coupled in on the inside, not the outside.
Taking a small step back: you know how a dipole antenna works, roughly (assume it's directly soldered on to the output of a differential amp): The amplifier causes harmonic voltage oscillations at the "beginning" of the two branches. Because "just a moment ago", the voltage at the neighboring points was still a bit different, that causes a harmonically changing current to flow in the dipole branches.
Current that flows along the length of the conductors causes a magnetic field, circling around the conductor. So, that changes to, harmonically, with the same frequency. Around the conductor. So, this has an effect on the environment of the dipole, not the dipole "metal sticks" themselves.
Changing magnetic field causes an electrical field, with a phase shift relative to the change of the magnetic field. That in turn causes a changing electric field, again in the environment: What we have is energy shifting from the magnetic to the electric field, and vice versa, alternatingly, and even more relevant to us: reaching further in space! We've caused an oscillating electromagnetic field, and what we got is a wave that radiates. Awesome; the physics for radio now stands! (Fast forward: billions of people watching cat videos on their phones. Who would have thought that differential equations involving electric and magnetic fields would have had that effect?)
So, now we do that, but we do it inside the coax. Same math/physics still works, but there's boundary coditions. These are:
- The center conductor is made of something very well-conducting. That means there cannot be an electrical field point "radially" inside the center conductor – electrical fields within a perfect conductor don't exist (if they existed, they would instantly be cancelled by the current they cause to flow). What can exist are electrical fields parallel to the surface, both in "forward" direction along the length of the coax, and in circular directions "around" the conductor
- The outer conductor (the "shield", which is a bad name, but commonly used; it's not shielding the waveguide, it's half of the waveguide) is also made of something very well-conducting. So, again, no electric field "radially", but sure longitudinally (i.e., along the length) and on the inner surface "rotatingly".
This has a couple interesting consequences. Namely, these waves that propagate, they have electric fields, but these are not "free"; they need to fulfill an equation that says "hey, at these points, your radial component needs to be zero, no discussion". That's fine – there's still "solutions" that allow for these waves to exist and propagate in the space between the inner and outer conductor.
So, back to your question: Why no current on the outside?
Well, what we've ruled out so far is that the electrical field from the inside leaks to the outside – that's the second condition there. But what about the displacement currents that might flow on the outer conductor? (Remember, electrical field along the surface is allowed to exist!)
Wouldn't they also work, just as in a dipole, to make an antenna out of the outer conductor? I just explained how current flowing in a conductor makes a dipole an antenna!
The truth is that the current flows on the (inner) surface of the outer conductor, because that's the place "in touch" with the electrical fields parallel to the surface. Now, while that does induce magnetic fields, these cannot extend "into" the conductor, to reach the other, outer side. The reason is simple: you put a changing magnetic field perpendicularly through a conductor, it inducts a current that is opposed and cancels the field.
Hence, these fields, both the electrical and the magnetic field, in the components that matter to the outside (electrical: radially, magnetic: parallel to the surface) can't exist at any depth into the outer conductor. We call that depth skin depth.
That is, as long as the conductor is perfect, that skin depth is arbitrarily thin, and the thinnest foil can effectively separate the inside from the outside. (And that is exactly why you can see, when you cut the center conductor of a coax cable, that its surface is plated with copper, silver or some other high-conductivity material, and the core is often made from aluminium or steel for cost and/or mechanical strength reasons: Deeper than a couple micrometers, the conductivity doesn't matter anymore, all the juicy currents happen within a skin depth from the surface.)
Alas, nothing is perfect, and certainly the outer conductor of your coax is not a superconductor. That means some field does leak to the outside, and your coax is a leaky waveguide. But: We've got that under control, relatively well. (Surprisingly well, actually)
So, for all practical purposes, a waveguide operated within the recommended frequencies and powers will be more or less leak-free. When you start doing higher frequencies with higher powers, your waveguides need to get more expensive: the quality and thickness of your outer and inner conductors matter more, and these currents that flow on the surfaces really start, even in a well copper-coated-then-silver-plated conductor, to see a bit of ohmic losses, which heat up the material, and you add another dimension of problems to your waveguide. Great. (Also, the dielectric, the nonconducting material between inner and outer conductor, is a lossy thing, gets hot, and/or stops being as nice and linear. That's why high-power coaxes are often mostly air-filled instead. Small mechanical problem: without some rigid material between inner and outer conductor, how do you keep the inner conductor exactly in the center? That's when you stop doing coax and just do hollow waveguides instead.)