Instead of facts we have perspectives, none privileged over the others as truer or more nearly in accord with things as they are, although not for that reason all equal.
Lets start to look at it in small bits...
Instead of facts we have perspectives
Ok, so we have something. And what we have are not facts (things that are true) but instead what we have are perspectives (ways to look at things).
none privileged over the others as truer or more nearly in accord with things as they are
None privileged means that none of those things we have (perspectives) is better than another one.
Better how? Well, better in the sense that they are more true (truer) than another.
Or, better because they are closer to being (in accord with) the way that things are (in reality) - or simply said, because they appear to be more like the real world.
although not for that reason all equal
But wait! We should not think they are all equal (in merit) just because (for that reason) none of them is better!
So, the whole sentence comes to something like this:
We have ways of looking at things, not facts.
Of these ways of looking at things, there is not a single one that is better.
Not a single one is better based on being more true.
Not a single one is better because it looks more true.
But even though not a single one is better than the others, that does not mean they are all equal.
Are they equal or not? We cannot tell from this sentence!
The only thing we are told in this sentence is that what we are told does not mean that they are equal.
Maybe they are equal - but for different reasons.
Maybe they are not equal - and the given reasons do not make them so.
All the author want to tell us is that we should not draw the conclusion that they are equal just because we cannot tell which one is more true or closer to reality.