The higher dimensions referred to two concepts.

One concept was the Six-Fold-Realm, or the six dimensions that existed outside of the standard five where most corporeal beings dwelled. It was inhabited primarily by transcendental beings like the Eternals, the Chronovores and the Guardians of Time. (PROSE: The Quantum Archangel)

The other concept was defined as a greater reality co-existing with this universe.

These dimensions were undetectable by third, fourth or fifth dimensional life, or even by a TARDIS. Experimentation about them had been banned on Gallifrey by the Dimensional Ethics Committee. Matter in this universe exposed to the higher dimensions was unpredictably altered, whereas living beings underwent psychological and physiological changes, and were usually driven insane. Some lifeforms, however, like the Old Ones were able to perceive the higher dimensions with less injury.

Ashkellia was built millennia in the past so the Old Ones could explore the higher dimensions. They opened a breach, but the results were disastrous. One of their number, Valdemar, sealed the breach and itself on the other side of it. (PROSE: Tomb of Valdemar)