How can I politely express that "I have understood"?

When my professor instructs me during his/her office hour, I may simply show my understanding by "Got it" or "I see". But I wonder how to say that politely and professionally in written English, especially in an email.

Typically when writing a polite and formal message or email; a longer sentence will sound more courteous and gracious than a two word reply.

There are innumerable ways of writing a polite acknowledgement, or a thank-you note. Change the adjectives with ones that you feel more comfortable with.

You can use the verb, grasp, instead of understand to say you comprehended something fully, and you won't easily forget or be confused by it again.

• Thank you so much for your clear and unambiguous explanations on [topic]. They have helped me tremendously.

• I really appreciated your explanations in the last session I attended, they helped me to grasp/understand [topic] and now I feel much more confident. Thank you.

What's wrong with "I understand." or simply "Understood."?

These are perfectly professional but also not stuffy.

Often the best way to indicate that you have understood is to summarize in one sentence what your understanding will lead you to do next.

e.g., "I understand. I will get back to you with the revisions you requested by Monday."

or

"I see. I will make sure to include or discuss your idea that .... in my next paper."

As in written form, there is no specific way to display politeness, the choice of words is all that can convey mood. Though, there are always ways to be more professional.

In an email, you could say "Thanks for the help, I have understood the subject."

This sounds polite (with "thanks" being used) and professional.