When an antenna is resonant, the impedance seen at the feed point is purely resistive and has no reactance.
What are the physical properties of an antenna that actually make it resonant and how do these properties relate to the frequency at which resonance occurs ?
The antenna has inductance and capacitance, which is the same thing a tank circuit has. A current traveling along the antenna creates a magnetic field; the field and the current interact with each other, thus it will be subject to inductance. The charges that accumulate on the conductors creates an electrostatic field that causes them to interact with each other (or with conductive objects nearby). The conductors thus behave like the plates of a capacitor. The combination is resonant at the frequency where the inductive and capacitive reactances are equal.
A resonant antenna produces feedback that is in phase with the applied signal.
For a more intuitive example, consider a mechanically resonant structure, like a tuning fork. If the fork is struck, a mechanical wave travels through the fork's structure. Some of the energy in this wave is lost as sound to the air, but most of that kinetic energy eventually propagates back to the point where the fork was initially struck.
The time it takes for this to happen determines the resonant frequency of the tuning fork and the note you'll hear when its struck.
If you were to repeatedly strike the tuning fork at this frequency, you'd find it would be "softer" than it would be at other frequency because right at the instant you're striking it, the fork is vibrating away from the striker. That is, the feedback is in phase with the applied signal.
Resonant antennas are analogous for electromagnetic waves. Consider a resonant dipole: each leg is tuned to be electrically a quarter wavelength long. The time it takes for the wavefront to travel from the feedpoint to the end is a delay of 90 degrees, and back from the end to the feedback another 90 degrees for 180 degrees in total. The time it takes this to happen is the time it takes the phase of the driving signal to move 180 degrees.
Because the feedback from reflections within the antenna is in phase with the applied signal at the feedpoint, the antenna is resonant.
Most antennas used in amateur radio are not proper resonant devices or resonators. The terminology "resonant" applied to the feedpoint impedance is purely a jargon of convenience (and very persistent) just to mean vanishing reactance component of the feedpoint impedance, not terminology representing the proper concept. Please make that point clear when you think about the physics of antennas, or one will mix up incompatible concepts and lead to erroneous statements.
Most antennas are standing wave devices, where the standing waves are formed on the antenna element(s), and therefore the feedpoint impedance and other behavior vary with the excitation frequency and other parameters. But these are not resonators that store and release energy in every cycle of excitation power. The antennas are made to couple with the fields (radiate and receive power), not store energy.
There are just a few antennas that are traveling wave or forward wave antennas, where reflection or standing wave is absent. Those include Beverage, rhombic, and axial helix. As you imagine, those antennas have a flat feedpoint impedance over a wide range of frequencies, but they perform well for certain applications.
There are a few antennas that are proper resonant devices. Loopstick (ferrite bar) antennas in MW broadcast receivers are best described as tuned resonator circuits that couple with the fields. Tuned small loop antennas (such as the so-called "magnetic loop," another loosely used terminology) can be understood in multiple ways, including as a tuned resonator.
