word that means "to understand deeply on an intellectual level"

Profound insight

The word "profound" is usually used to mean "deeply" in cases of understanding/knowledge. "Insight" has stronger connotations of experience or direct knowledge rather than intellectual or theoretical understanding.

Sorry, can't come up with a single word.

I would choose

comprehend
1. to grasp the nature, significance, or meaning of

and

empathize
1. to have the same feelings as another person : to feel empathy for someone

Examples

• Someone is explaining something in depth and at length, then after a long segment without stops, he looks to his audience for feedback. A correct response would be a reply "I comprehend".

• A dear friend is explaining the circumstances of an unfortunate disaster that is very similar to one you have also experiences. A correct response would be "I empathize".

Pragmatically

For both examples above they are "correct" responses, but perhaps not the best options. In general speech, people tend to use the word "understand" in both circumstances. In reference to the circumstances, however, "understand" is lacking and the other words may be used to ensure communication is effective.

Modified Examples

• "I found that I was able to comprehend the lecture and that was apparent in my test scores."
• "I empathize with children from fractured homes; my parents were also divorced."

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In vernacular use the phrase "get it" is synonymous with "understand". However, it has also been extended to circumstances where you may comprehend everything that is said, but do not find it valuable or find that it misses its mark (i.e. a joke that's not funny).

Viscerally - is to relate to deep inward feelings rather than to the intellect. I am seeking a word for the "intellectual equivalent" of that - but visceral provides half of your answer!