The simple explanation is that oscillation occurs when you have too much gain compared to the reverse isolation (S12). If S21 is too high compared to S12, then you might have a situation where the signal gets amplified, then reflected from the output load, then attenuated through the isolation, then reflected back from the input load with higher amplitude, which causes oscillation.
The only way to increase stability is to waste power somehow to decrease the gain of the system. This is why only resistors can be used to improve stability. Changing the matching network is useless when it comes to improving stability (assuming the components are ideal) because ideal inductors and capacitors do not dissipate power.
In the circuit shown here, at low frequencies the resistor R3 affects the gain a lot. This is because the inductor looks like a short and the capacitor looks like an open, so power is dissipated in the resistor. At high frequencies, the capacitor dominates over the resistor since it basically shorts it out. This makes the resistor waste less power. The point of L2 and C4 is basically to decrease gain at low frequencies, and keep the gain relatively similar at higher frequencies. This is useful when the transistor is unstable at low frequency, but stable at high frequency. If you just used a resistor, then the gain would drop at all frequencies, but this circuit would keep the gain similar at high frequency while improving stability at low frequency.
Another useful trick is to put a resistor in series with a quarter wave transmission line and stub. At the frequency of interest, the quarter wave line and stub look like an open, so the circuit basically looks like a resistor with one end open. Therefore the gain is not affected at all at this frequency. At other frequencies, the stub and line no longer look like an open circuit, so the resistor will waste power and therefore stability would improve at all other frequencies.