Injection and acceleration of the background plasma electrons in laser wakefield accelerators (LWFA) operated in the blowout (‘bubble’) regime are analysed. Using a model of a slowly expanding spherical plasma bubble propagating with an ultra-relativistic speed, we derive a sufficient condition for the electron injection: the change in the electron’s Hamiltonian in the co-moving with the bubble reference frame must exceed its rest mass energy m_{e}c^2. We demonstrate the existence of the minimal expansion rate of the bubble needed for electron injection. We demonstrate that if the bubble’s expansion is followed by its stabilization or contraction, then a quasi-monoenergetic electron beam can be produced owing to the phase space rotation of the beam inside the bubble. Using particle-in-cell simulations, we verify that the temporal expansion of the bubble is indeed the dominant effect responsible for electron self-injection and trapping in the rarefied plasmas relevant to LWFA with petawatt-class lasers.