You were doing well up to the end. Free space doesn't resonate. It just presents an impedance. All we are trying to do with antennas is to couple energy into free space. It happens to work very well with a resonant length of wire, but we can do it with a much longer or shorter length of wire, or we can do it with a loop of wire.

Into the capacitance of the dipole. Why is that so hard to imagine? Current flows when you connect a battery to a capacitor, even though it seems it has nowhere to flow.

Simply, the feedpoint impedance of your antenna must be equal to the characteristic impedance of your feedline. If they are not equal, some power is reflected back down the feedline, resulting in a standing wave. This will probably make your transmitter unhappy, as this power must now be sent back to the antenna again. There are a number of ways you can make the impedances equal, such as pi networks, l networks, transformers... the list goes on and on, and a discussion of impedance matching could easily fill several books.

Make no mistake, the capacitance does exist, and it is not negligible, as Neil said above. You can easily measure it with a DMM, if you were inclined to do so. Two parallel plates aren't the only way to generate capacitance. It seems counterintuitive, I know, but two wires running in opposite directions *will* have a real capacitance between them, far more than surface areas would lead you to suspect. For example, a short dipole, about 1m on each leg, 1mm in diameter, has a capacitance of about 20pF. Increasing the diameter to 10mm increases this to about 30pF, whereas increasing the length to 10m increases the capacitance to about 135pF. This is a fairly decent amount of capacitance.

In order for an antenna to radiate, the basic requirement is this: alternating current must flow a distance through a conductor. The more current and the longer the conductor, the more the antenna radiates. Let's look at your halfwave dipole example a little more closely. Imagine what happens along the length of your dipole antenna, if a 10V sine wave is applied at the feed point. When one end is at 10V, the other end of the dipole is at 0V. They, of course, switch off rapidly, setting up a standing wave. The ends of the antenna are swinging up and down in potential, while at the feedpoint, there is almost no change in voltage at all. These large voltage potentials cause RF current to flow through the antenna. Current flows from one half of the antenna to the other, reaching maximum current right at the feedpoint. This current does the job of radiating. The total amount of current that actually flows is determined by your antenna's feedpoint impedance. In this case, roughly 73 ohms, purely resistive.