How does the Colpitts Oscillator reach a loop gain of 1?
How does the Colpitts oscillator reach the loop gain \$A_V∗B\$ of 1?

Maybe it's best to use a simulator to show where the gain becomes limited. Here's the “basic” circuit I used and note, that in the first instant, I didn't connect the emitter capacitor C4: -

Note the waveforms; blue is Vout and red is Ve (emitter): -

They &quot;collide&quot; at about 3.1 volts and this prevents any serious increase in output amplitude. In other words, the “basic” common-emitter Colpitts oscillator will always tend to have a significant sinewave distortion.
This Colpitts CE website is now available should more detail be required.
Back to the answer.... It's the same story if I connect C4: -

This time there is a little more output amplitude but again, troughs in Vout collide with Ve and cause asymmetrical clipping. This limits the amplification of the circuit and results in amplitude stability albeit with distortion.

And why this is only possible to LC oscillators and not to RC
oscillators for example a Wien-Bridge oscillator

A Wien bridge oscillator will increase its output amplitude until it &quot;crashes&quot; into one of the power rails and therefore attains gain stability through distortion (just as the Colpitts example does).

Some Math
As for the theory behind the frequency of oscillation, you have to regard C1, C2, L and the effective output resistance of the collector acting as a third order network delivering a phase shift of 180 degrees: -

$$\dfrac{V_{OUT}}{V_X} = \dfrac{1}{1+s^2LC_2}\text{  ....take note for later}$$
And, the impedance of C1, L and C2 (\$Z_X\$) is: -
$$Z_X = \dfrac{1+s^2LC_2}{s^3LC_1C_2+s(C_1+C_2)}$$
Therefore (and with a couple of lines of math skipped): -
$$\dfrac{V_X}{V_{IN}} = \dfrac{1 + s^2LC_2}{s^3LC_1C_2R + s^2LC_2 + sR(C_1+C_2) +1}$$
Dividing the transfer functions to get rid of \$V_X\$ yields: -
$$\dfrac{V_{OUT}}{V_{IN}} = \dfrac{1}{s^3LC_1C_2R + s^2LC_2 + sR(C_1+C_2) +1}$$
Noting that for the overall TF to only have a resistive transfer function, the imaginary parts in the denominator cancel to zero hence: -
$$-j\omega^3 LC_1C_2R + j\omega R(C_1+C_2) = 0$$
Therefore R (and of course j) cancel on both sides and, the TF reduces to: -
$$\omega = \sqrt{\dfrac{C_1 +C_2}{LC_1C_2}} = \sqrt{\dfrac{1}{LC_2}+\dfrac{1}{LC_1}}$$
This informs us that the oscillation-frequency feedback is not at the amplitude resonance of L and C2. The oscillation point is on the slope of L and C2 i.e. off amplitude-resonance. You might notice that &quot;R&quot; falls out of the equation and that is also covered a little lower down.
Going back to the main transfer equation (with imaginary parts in denominator at zero) we have: -
$$\dfrac{V_{OUT}}{V_{IN}} = \dfrac{1}{1-\omega^2 LC_2}$$
And, if we plug-in the oscillation frequency (\$\omega\$) we get: -
$$\dfrac{V_{OUT}}{V_{IN}} = \dfrac{1}{1 - \dfrac{C_1+C_2}{LC_1 C_2}\cdot LC_2}$$
And drilling down we find that: -
$$\dfrac{V_{OUT}}{V_{IN}} = -\dfrac{C_1}{C_2}$$
Hence, if C1 equals C2 we get a unity amplitude transfer function for R, L, C1 and C2. If we did a simulation of the third order filter we would see that the value of &quot;R&quot; does not affect the phase angle nor the amplitude response at the oscillation frequency of 2.2508 MHz: -

Note that the oscillation frequency is not quite at the amplitude resonance either. It gets pretty indistinguishable at high values of &quot;R&quot; of course.
And, if you did the math, 2.2508 MHz = \$\sqrt{\dfrac{1}{LC_2}+\dfrac{1}{LC_1}}\$

An improvement
Because the common-emitter Colpitts oscillator has plenty of gain it's very likely (in examples on the web) that there will be high distortion levels. I would never consider running one of these circuits with an emitter capacitor because the gain will be too high and asymmetrical clipping will result. In fact, because I have the simulator open, I'd do this to get a decent sinewave: -

Notice the back-to-back diodes (1N4148) that clamp the signal to +/1.4 volts (ish) and notice that the feedback comes via a 33 pF capacitor. I've also reduced the emitter resistor to 470 ohm to allow a tad more headroom and, lowered R3 to 3k3 to lower the bias point: -

That's a 6 volt p-p output and very little sinewave distortion. It's all about providing just enough gain to get the circuit started and having sufficient and progressive gain reduction (as signals rise) to get amplitude stability without too much distortion.
I would probably get rid of the collector inductor and replace it with a 1k8 resistor in many applications: -

The sine wave amplitude is reduced (as expected) but purity still looks half decent: -

And finally, remember that most circuits on the internet that describe oscillators are very basic in nature and, in most cases, to make a decent practical oscillator requires a little bit of design refinement. After all, if a circuit is described as a sine wave oscillator, you’d probably expect it to produce no visible distortion on an oscilloscope if you bread boarded it.
It’s a shame that many sites don’t go that extra mile.