Word(s) meaning "to understand deeply"

Solution 1:

grok

The word that comes to mind is grok. This term originally comes from Robert Heinlein’s 1961 science-fiction novel A Stranger in a Strange Land, but it has since entered the popular culture, and Merriam-Webster defines it as “to understand profoundly and intuitively”.

Solution 2:

To grasp.

Comprehend fully:
‘the press failed to grasp the significance of what had happened’

Reference:
https://en.oxforddictionaries.com/definition/grasp

Solution 3:

One option is to use a metaphor of depth:

Sally had fathomed the plot.

The professor had spent twenty years penetrating the arcana of Phoenician religion.

We got to the bottom of it.

Metaphors for 'absorbing' or 'eating' could work, too:

I absorbed the course material.

Wendy has assimilated the entire book.

He digested the pamphlet.

They took it all in.

Solution 4:

If it is a concept that is being understood, then comprehend may work.

to grasp the nature, significance, or meaning of. Ex: "unable to comprehend what has happened"

Usage: "After years of study, he managed to comprehend quantum physics. His wife, not so much."