When trying to figure out whether or not a given reaction will exist at a given support, it's worth remembering what a reaction actually is.
A reaction is the means by which the support resists the movement of the beam at that point. Force reactions resist the beam's attempts to deflect up-down or left-right at the support. Moment reactions resist the beam's attempts to rotate at the support.
So, let's assume there's no bending reaction at the beam's extremities. In that case, we'll be simply dealing with a simply-supported beam with pinned supports at the ends. These pinned supports will generate vertical forces which resist the beam's deflection at those points.
But how would the beam deflect in this case? Well, it'd obviously be shaped like a parabola (well, it'd actually be a cubic function, but it looks parabola-ish).
But that means there was rotation at the extremities: the beam was previously horizontal at the supports, but now it's tilted.
But the definition of a fixed support is that it doesn't allow for rotations at that point.
So what does the fixed support do to stop that rotation? Well, it applies a concentrated bending moment to the beam, rotating it back to horizontal:
Figures created with Ftool, a free structural analysis program.