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The existence of extra dimensions is explained using the Calabi-Yau manifold, in which all the intrinsic properties of elementary particles are hidden.
Credit: A Hanson.

Explained, mind you. I now have no excuse for not understanding it.

In theory, if you could master the fifth and sixth dimension, you could travel back in time or go to different futures.

What is the nature of those fifth and sixth dimensions that would give you privileged access to the fourth? If not spacial, then what is it?
What does "master" them mean exactly? Understand them or ability to observe or manipulate extra dimensional objects?
This hypothesis that our 3space+1time dimensional universe is inside a much larger universe with more dimensions is far from new.
It's not the same as the theory that our universe may have additional dimensions that are subatomic in size.
Inexplicably, that too happens to be 10 or 11 dimensions according to some, maybe the author got it mixed up.
No, this appears to be the realm where alternate universes and histories reside and where each dimension somehow unlocks more "mysteries".

I call hooie on this.

The math for N-dimensional space is already well known.
We have at present no way of observing any extra dimensions.
This is pure speculation. Fun reading but speculation nonetheless.