You can read a review of the paper in Zentralblatt, it contains a short description in German.
The review on MathSciNet is a bit more extensive (but requires a subscription). There is indeed the condition of $n+1$-to-one from a zero-dimensional compact space onto the space itself. There is also a condition on `gratings' (not defined, but presumably special covers of the space).
In What is a non-metrizable analog of metrizable compacta? (Part I) Pasynkov defines a class of compacta where the dimensions coincide.