Interesting question. Note that the power transfer is $$P=\frac{1}{\tau}\int_0^\tau F(t)v(t)\,\mathrm dt$$
where $F(t)$ is the applied force, $v(t)$ is the velocity of the oscillation, and $\tau$ is the length of the period of the oscillation (this is just a simple extension of the usual rule for work, $w=Fd$, to time-varying systems).
The usual forced oscillator
$$m x''(t)=-kx(t)-\gamma x'(t)+A\sin(\omega t)$$
has steady-state solution
$$x(t)=\frac{A }{\sqrt{\left(k-m\omega^2\right)^2+\gamma ^2 \omega ^2}}\sin \left(\omega t+\tan ^{-1}\left(k-m \omega ^2,-\gamma \omega \right)
\right).$$
Since the user-applied force is $F(t)=A\sin(\omega t)$, the velocity is $v(t)=x'(t)$ and $\tau=2\pi/\omega$, we obtain
\begin{align}P&=\frac{\omega}{2\pi}\int_0^{2\pi/\omega} A\sin(\omega t)\frac{A \omega \cos \left(\omega t+\tan ^{-1}\left(k-m \omega ^2,-\gamma \omega
\right)\right)}{\sqrt{\left(k-m\omega^2\right)^2+\gamma ^2 \omega ^2}}\,\mathrm dt\\&=\frac{\frac{1}{2}A^2 \gamma \omega ^2}{\left(k-m \omega ^2\right)^2+\gamma ^2 \omega ^2}\,.\end{align}
Solving $$\frac{\mathrm dP}{\mathrm d\omega}=0$$ for $\omega$ yields
$$\omega=\sqrt{\frac{k}{m}}$$
which is exactly the natural frequency.
Thus, somewhat paradoxically, maximum power transfer occurs when the forcing is at the natural frequency, even though the vibrational amplitude is not maximum.
(Minor note: I am using the two-argument arctangent function $\tan^{-1}(a,b)$ as it is defined in Mathematica, ArcTan[a,b]).
Since you want to answer a small child, I'd just start with a weight on a spring, preferably with a physical demo at hand. Show how it bounces at a specific rate regardless of how you initially stretch it out. Then maybe show a different resonant rate when you change the weight, or length of spring. That takes care of the "natural resonance frequency" part.
Then, maybe a bit more tricky: gently tap the weight (vertically) at the resonance rate and show the amplitude growing. Then tap at some other rate and observe the amplitude collapsing or going pseudorandom.
Best Answer
Re question 1: when you learn this stuff in school you usually simplify the system by modelling it as a simple harmonic oscillator so the amplitude of the system will be given by some equation like:
$$ A(t) = A_0 e^{i\omega_0 t} $$
where $\omega_0$ is the natural frequency of oscillation. Typically you study what happens if you apply a force that also varies sinusoidally with time so:
$$ F(t) = F_0 e^{i\omega t} $$
where the frequency of the applied force, $\omega$, is not necessarily the same as the natural frequency of the oscillator, $\omega_0$. This is what your teacher means by saying that the force has a frequency - they mean the frequency $\omega$. In your teacher's example of a swing the swing has some natural frequency. If you are applying a force periodically, i.e. pushing on the swing in a repetitive way, then the force you apply also varies with time (though it is more like a square wave than a sine wave). The amplitude of the swing is greatest when the frequency with which you push the swing matches the natural frequency of the swing.
Re question 2: when you start learning this stuff you typically start with an undamped simple harmonic oscillator, i.e. the oscillator doesn't lose any energy. If you solve the equations of motion you find that the amplitude goes to infinity when the frequency of the driving force $\omega$ is equal to the natural frequency $\omega_0$. This is because you're putting energy in but the oscillator doesn't lose any energy so the energy just keeps growing.
A real oscillator like a swing loses energy through friction, and we call it a damped harmonic oscillator. The rate at which the oscillator loses energy is related to its amplitude, so as you push your system (the swing in this case) the amplitude increases until the rate of energy loss matches the rate you're putting energy in. So the harder you push your system the more the swing will move. In principle there is no maximum amplitude, though in real life there obviously is since at some point the swing will go over the top and start revolving instead of swinging to and fro. A swing isn't a simple harmonic oscillator! It's only approximately simple harmonic for small swing amplitudes.
Re question 3: Most objects will have a range of resonant frequencies called normal modes. However there are usually many normal modes and the frequencies of these modes are related to the object's shape in a complicated way. The Wikipedia article gives some examples of normal modes, or do a YouTube search for "normal modes" to find loads of videos on the subject - some really impressive!