This week we read about the famous story of how "Pesach Sheni" (2nd Passover) came to be.

The Torah tells us the story of how Am Yisrael were celebrating their first Pesach after the Exodus, during which several people were Tam'e (impure), and therefore were not able to celebrate Pesach with all of Am Yisrael.

Feeling left out, they approached Moshe and told him that they too wanted to celebrate Pesach on time.
Hashem then instructs Moshe the rules of Pesach Sheni, that shall be celebrated for generations to come

When reading this story, 2 main questions come to mind:
1. Why were these people feeling left out only by Pesach? What about the other Holidays? If a person is impure, then they can not celebrate those Holidays too and participate in the Korbanot of those Holidays too - so why here by Pesach, do we have a Pesach Sheni?
2. But even more than that, the people complained that they wanted to celebrate it במועדו - "in its time". Pesach Sheni is not "in its time" - so how does this solve their problem?

What exactly is going here?
What is Pesach Sheni all about?