How do I explain to a prof who is not happy with my actions which were simply a mistake due to miscommunication? I had emailed a professor for a summer internship opportunity. He had accepted me and aksed me write a proposal for applying through a fellowship.

I replied back putting down some points for my proposal. I got no reply from his side. I emailed him again after 2 days and continued to do so for the next 3 weeks. Afterwards, due to no response, I believed he was not interested in me due to no email response and stopped mailing.

But now, after the deadline is over, I found out from another source that he was not happy with my behaviour for not applying for the fellowship. 

How do I convince him that this was a honest mistake and I did not intend to be rude and am still interested in working under his guidance?
Agree with Dave, as supervisor/student relationship is like a marriage, communication is essential for it to remain healthy.

However also note that, since you have emailed him quite a number of times and he never responded with any feedback, he did not want to help you, and hence he might not be the kind of person you want to work with. There is nothing worse than a bad supervisor in academia.