1. It tells us that it is most probably a man that is speaking.
2. The first 僕 is about who is reminiscing (覚えている) as the second one help describes what he is reminiscing about… About the day (someone) was born.
3. 生まれた日 Day (subject) was born (Day of birth)
4. I am reminiscing about the day I was born…

These are my guesses… I chose reminiscing instead of simpler verb remembering because of the different feeling that is attached to it as not simply remembering in the sense of not forgetting but also implying thinking about this day with regret or melancholy…