The flat portion is when the diodes are conducting as others have stated. If the capacitors are increased in value, this flat region will be shorter as more energy will be stored in them.
The spike is the inductive "kick back" when the diodes stop conducting and there is no longer a load on the transformer. The magnetic field collapses and produces a reverse voltage.
This "old school" method of linear rectification, which is in a lot of old devices, creates harmonics on the input power lines that makes the power companies unhappy especially in Europe.
Many commercial installations have a limit on harmonics on the power line. Not so for residential, yet.
Randy