How does this BJT oscillator work step by step?
Well, I think that a step-by-step description is not a suitable method for describing the working principle of a harmonic oscillator - more than that: It is not possible and/or does not improve the understanding. This applies to all linear/harmonic oscillators.

Hence, it is best and sufficient to show if and how the oscillation criterion (Barkhausen) will be fulfilled. This criterion requires that we have positive feedback (loop gain larger than "1") with a phase shift of 360 deg (0 deg) at ONE SINGLE frequency fo only (the desired oscillation frequency).
Let us see if the criterion is met:

1.) If both nodes labelled "Vo" are connected we have the chance to provide positive feedback if the phase shift caused by the capacitors allows a phase shift of 0 deg. (Both transistor stages are inverting and do not add additional phase shifts).

2.) Both capacitors (C1 and C2) - together with the corresponding resistive parts form a bandpass with a zero deg phase shift between both stages for one single frequency fo. It is, in principle, a C-R-R-C ladder bandpass structure:

3.) C2 realizes - together with the resistive input of Q1 - a highpass function and C1 is responsible for a lowpass function with a higher corner frequency than for the highpass (C1 is smaller than C2). 

4.) This lowpass function comes from the fact that with rising frequencies the capacitor C1 causes an increase of negative feedback for the Q1 stage and, thus, a decreasing gain of this stage. 

5.) The total gain of the loop (determined by R1 and R5) must be large enough to ensure a loop gain which is larger than unity at f=fo.