Let me try to say in intuitive terms what I think is really going on here.
The problem we face is as old as the problem of other minds, or intersubjectivity, or even commensurability, and it naturally involves a whole slew of other old problems — reality and appearance, or reality and representation, not to mention the one and the many. One way to sum up the question might be “conditions on the possibility of a mutually objective world”.
Working on what oftentimes seems like the tenuous assumption that there really is a real world causing the impressions in my mind and the impressions in yours — more generally speaking, that there really is a real world impressing itself in systematic measures on every frame of reference — we find ourselves pressed to give an account of the hypothetical unity beneath the manifest diversity — and how it is possible to discover the former in the latter.
Manifold theory proposes one type of solution to that host of problems.