How do I demonstrate that I read and understood a professor's paper to the professor? This is a question that has been in my mind for a long time. Now I am a M.Sc. first year student  in mathematics and planning to apply for graduate school next year in Europe. But for applying to graduate schools, it is written that first I have to write an email to a supervisor along with a idea of a research project and I have to convince him that I am well prepared to work under him as a PhD student. Also,  it is recommended that you should study the papers of professors to whom you are applying.
Now, I am interested in a topic. My college does not offer that topic. So, I have learned the topics on my own. Also, at this moment I do not participate in an internship program because of this pandemic situation all over the world. That is why I have decided that I will read some papers of the professor to convince him that I am well prepared to work under him.
My question is as follows: How do I demonstrate that I read and understood a professor's paper to the professor?
Please advise me. Thanking in advance.
That is why I have decided that I will read some papers of the professors to convince him that I am well prepared to work under him.

I disagree with this advice. I would rather see applicants (especially those with an MS) demonstrate their ability to do research, usually through a publication or a strong thesis. As long as their topic is somewhat related to my research, that would be a stronger demonstration of their potential to succeed in a PhD program. Being familiar with specific papers that I have written is not necessary or expected (you can always read such papers if you are accepted).
If your only option is to study the papers of professors, my advice is to implement the methods discussed in those papers numerically, i.e. repeating the results, or possibly even conducting some trivial yet new extensions (like repeating the experiments on a different dataset). That would be far more impressive than simply reading a paper (where your depth of understanding is not clear).