SAW resonator oscillator operating principle
C1, C3 and the SAW filter from a pi filter that in conjunction with the output impedance of the T1's collector produce a perfect 180° phase shift from T1's collector to base at one particular frequency. Given that T1 is configured as a common emitter amplifier, it inherently produces signal inversion (also 180° of phase shift) and therefore, together, you get 360° of phase shift at one particular frequency and the circuit oscillates when T2 (the modulation transistor) is activated.
360° of phase shift is of course the same as 0° of phase shift and that is part of the criteria needed to make a successful oscillator.
The SAW filter very much behaves like a proper crystal and here is a full tear-down of how a crystal oscillator works. Your circuit is quite similar to the Pierce Oscillator that is commonly used in crystal oscillators. It's basically a spin-off from the Colpitts Oscillator.