Well, I'm not exactly sure what Riemann had in mind, but if I had to guess it has to do with the principal of locality. Since Riemann appears to be explaining by way of analogy, by appealing to a physical (or perhaps metaphysical) concept like an instance (or maybe a moment), I'll continue with the analogy by appealing to physical concepts.
Almost all of physics can be broken into two different studies: that of dynamics and that of kinematics. Dynamics is the study of objects under the influence of causes, such as a particle moving in the presence of an electromagnetic field, or the motion of a ball after it has been pushed. On the other hand kinematics seeks to study the motion of objects without reference to causes.
So what could be meant by instance, is really the moment at which something affects another thing. The Euclidean model is very good at describing the dynamics of an event. Generally, one can describe the dynamics by imposing a coordinate system and describing the position of using the coordinates. One can then describe the motion of the object in these coordinates after some cause affects the object.
On the other hand, in the presence of nontrivial kinematics, the full effect of the cause is more difficult to describe. The idea is that motion of the object itself is subject to some nontrivial geometry independent of the cause.
Think of kinematics as being described by manifolds: the bare geometry on which points live; think of dynamics as modelling motion under the influence of causes without reference to this geometry. In this analogy, what manifolds do is allow us to describe both the kinematics and dynamics of an event locally. A local coordinate chart tells us how the dynamics at each point changes, as we vary the point in a continuous fashion...so sort of a continuous transition of an instance.
Of course, this is all guesswork. Since I have no idea what Riemann really meant by that statement.