While this video is a little dry (it was filmed during a university lecture), it is helpful in grasping the implications of the Borde-Vilenkin-Guth (BVG) Theorem. Notice at about the 3 minute mark the BVG Theorem explicitly stated. In layman’s terms, Dr. Vilenkin is saying that a time traveler moving forward in time notices that her travel relative to the expansion of particles in spacetime slows down (which is red shift) and that, if she were to move backwards in time (towards the Singularity), it would speed up (blue shift). Since nothing travels faster than the speed of light, she cannot travel infinitely backwards in time because, at some point, she will reach the speed of light. Therefore, the expansion of spacetime itself had a beginning in the finite past.
Notice also, at around 5:45, Vilenkin’s comments on other inflationay models that attempt to evade the BVG Theorem.