I am looking for a PhD position. Sometimes professors give me information about their projects via emails. How can I thank them for their explanation in a polite way, show my interest and impress them?
Natalie Hogg's answer is spot-on. I want to address a further aspect of the question:
How can I thank them for their explanation in a polite way, show my interest and impress them?
While impressing your potential advisor is certainly a desirable goal, the chances of impressing a professor with a thank-you e-mail are close to zero. A professor is more likely to by impressed by an applicant if the application itself is impressive, for example, because it includes convincing arguments why your particular skills and experiences will help their work group produce good research.