This is a tricky question of cause and effect.
Is there low pressure because the flow accelerates or is the flow accelerating because there is low pressure? If you ask me, both, both are cause and effect at the same time. If one of them fails, so does the other.
Now I'll try to explain it in a more mechanical "what goes where why" way. I like masses pushing masses more than I like Bernoulli, even if in the end they describe the same phenomenon.
To start with, there is a motor: The large force driving it all is the vacuum hole that the car creates behind itself as it punches through the air. The diffuser is an elegant way of moving this partial vacuum from behind the car, where it causes drag, to below the car, where it creates downforce. So the vacuum cleaner analogy is not at all out of place here.
Now to your two questions:
Why does air from the sides get attracted to the diffuser?
Air from everywhere rushes in to fill in the hole behind the car. The diffuser makes sure that some of it comes all the way from in front of the floor, but also air from the sides (and from above and from far behind the car) is trying to rush in. Skirts in the 80s were introduced to counter this effect. Today's cars live with it, but it is detrimental to downforce.
Why is there a speed peak / pressure low at the kink or if it be, the convex transition at the beginning of the diffuser?
You can think of it as a Coanda effect. Air is moving below the floor towards the back of the car (because the diffusser is giving it a clean path to the vacuum cleaner engine). Then it reaches the kink. This mass of air, if it were to continue to move straight, as it entrails air around it, would create a vacuum right behind the kink point. That of course won't happen (to any large extent), and any air available will move in to fill that partial vacuum that is being created right past the kink. This is the very same air trying to create the vacuum, which then a) bends upwards to fill this partial vacuum b) accelerates against a partial vacuum to do so (there is less air opposing it there than air pushing behind it to go there). Think of a gas expanding against a perfect vacuum. It accelerates as it goes there because nothing is pushing back.
So again, the Coanda effect is happening because there is low pressure there, and there is low pressure there because there is Coanda going on.
Confusing? To me the Bernoulli version is confusing.
Wind turbines are cool, elegant and magnificent. TANSTAAFL!