How does the high voltage get to the ends?
In short, the electromagnetic field set up around the antenna tend to shove a high density of electrons to one end or the other at different parts of the RF cycle.
In the middle of the antenna, the electrons can easily travel further down the antenna, but when they reach the antenna ends and the electric field starts getting strong, they pile up there.
As far as the DANGER goes, something you need to know is that RF tends to travel on the outside of your body when you come in contact with it. The conductivity of your body tends to exclude currents from traveling straight through you.
This is called skin effect, but it's named for the effect in other conductors, not the effect on people :-) Electromagnetic fields can't penetrate far into a good conductor. A perfect conductor would exclude them entirely. So the current density in a conductor decays exponentially as you go in from the outside.
This means that bridging hundreds of volts of RF is much less serious (i.e. not fatal), where bridging several hundred volts of 60Hz or DC can kill you dead, while the same RF voltage will probably just hurt and maybe give you a nasty black spot where you get a "RF burn." I read some cases of people who came into contact with quite high voltage RF and got a serious full body RF burn with some lingering health effects... but there don't seem to be many cases of fatal RF electrocution even from very high power transmitters. You don't want to mess around but it's not like you're playing with 1kVDC...
Is the danger as great at the center of a dipole?
Nope, not even close. I know it's weird for the voltage between the two dipole halves to depend on where you are on the dipole, but it's because of the way the electromagnetic field is set up around the antenna. Near the middle, there's a lot of magnetic field encircling the wire. At the ends, there's a strong electric field that "starts" on one end and goes to the other, if you like thinking about field lines.
Of course, at EVERY point on the dipole there's some electric field ending on the wire and some magnetic field encircling it, but at the middle, there's a lot more magnetic field from a high current flowing in the wire, and much weaker electric field. So the voltage from one dipole half to the other at the feedpoint is low compared to the ends... a hundred volts peak (70.7V RMS) at 100W into a 50 ohm load.
It is true that coming into contact with antenna tips even at low power isn't nice. I got my first RF burn when I was a ham... running 4W of CB power into a half square I built. I touched the end of that and it really stung.
The nastiest RF burn I got was off a 10m matching network on a random wire I was adjusting with 30-40W or so. That sucker made a tiny hard black scorched spot on my skin that lasted for several weeks. The problem with high voltage RF or lower frequency, of course, is that it easily ARCS to you as you pull away. And at high power arcing to trees or whatever can pretty easily start fires.
So don't mess around, but also don't think that you can casually kill yourself with a low power transmitter's output and the voltage step up inherent in a dipole antenna. It's very high voltage, but it's not like working inside an amplifier or contacting high voltage power lines, just because the current can't get inside your body easily.
73
Dan