First of all some clarifications to make sure we are on the same page. The phrase "$A_\nu$ is local" is slightly misleading as it implies that A is only defined in some (small) neighborhood of the total manifold/space. This is not true. A is defined everywhere on the manifold. The correct expression is that "the theory is invariant under a local symmetry" ie that transformations of the form
do not change the physics. The above are known as local/gauge transformations and they are distinctly different from global transformations in which the function does not depend on the position, ie:
Now the statement about the Maxwell's equation mentioned, is that we can find a solution to it at any small neighborhood of the manifold, but when we try to patch together solutions in nearby neighborhoods to find a solution that will satisfy the equation everywhere, then this is impossible (for manifolds that have "holes").
The basic (physical) reason behind this is that for manifolds with no holes a field of the form $A=$constant is not physical and we can do a gauge transformation of the type described above to bring it to the form $A=0$ everywhere. However, for manifolds with holes (an example would be the circle) a field that is constant everywhere is physical, it cannot be gauged away and has observable consequences. This is another manifestation of the Aharonov–Bohm effect.
So in the example above, if the constant magnetic field along the circle is physical then what creates it? We can think of its source as an electric current flowing through a long wire passing through the center of the circle and being perpendicular to the plane of the circle, or equivalently we can think about it as having placed a magnetic monopole at the center of the circle. This simple example hopefully demonstrates why magnetic monopoles are needed to be introduced to justify global solutions of Maxwell's equaitons on a space with non-trivial topology (ie with holes).