There are several reasons:
1. "honeymoon period" with Congress in first 100 days
This is a period when there is less resistance to him. This is more pronounced when President and Congress are not united.
This was covered by research by Casey Dominguez, Department of Political Science
at UCB. There are two different papers he published on the topic:
2. A "honeymoon period" with the Press coverage.
Quoting from Dimonguez's paper above, in prior research (this is mentioned in both papers)
In their study of the press’ treatment of the president, Michael Baruch Grossman and Martha Joynt Kumar (1981, 259-265; 275-279) found that during the inaugural year, the press covered the president in greater volume and in a more positive light than later in the term. Their analysis showed that early coverage was favorable to presidents because reporters rely heavily on official White House briefings while they develop their own administration sources, and because reporters and editors believe announcements of policy proposals and profiles of new Members of the administration are inherently newsworthy. If conflicts or scandals emerge in the first few months of a presidency, the press will cover them in ways that might be perceived as negative toward the president, but barring major mis-steps, presidents can expect their press coverage to be neutral or positive.
(Unless you are Trump. All bets are off there, it's a hate-hate relationship, he even ticked off Fox News, for better variety :).
3. Also a "honeymoon period" with the public, partly caused by favorable press coverage.
Reasonable people will give you SOME benefit of the doubt before judging you on what you did.
Partially in response to that favorable press coverage, the president has been found to have a honeymoon with the public during his first few months in office. Approval ratings are stable and higher than average during their first few months. ... At the beginning of an administration, when polls ask whether respondents approve of the job the president is doing, partisans of the president respond by saying they approve of his performance, as do a sizeable number of independents and opponents who will change their minds once he actually builds a record on which to make judgments. Many others offer “no opinion”(Brody 1991, 28-44).
4. Better success with actual bills.
Research also shows that presidents have higher “success rates” when they take positions on bills before the Congress during their inaugural year than they do later in their terms (Dominguez 2005; Lockerbie, Borrelli and Hedger 1998).
Why 100 days specifically?
It's a nice very round number, perfect for a soundbite and a meme.
I wasn't able to find any conclusive proof, but following Wikipedia links, there appear to be two "scholarly" speculations:
Overall term and 100 days idea is claimed to have been originated with Elba-to-Waterloo period of Napoleon Bonaparte's return (source: Alter, "The Defining Moment: FDR's Hundred Days And the Triumph of Hope")
In politics, the claim is that FDR originated the term, by referring to (accidentally being exactly 100 day long) period of 73rd Congress session that created the New Deal. Ironically, that was NOT FDR's first 100 days as President.
So, why ~3-3.5 months?
Several likely reasons, all my speculation:
Less than 3 months, and you can't really get anything much done worth making into a memorable list.
Partly, because you need to assemble the team of higher ups AND hire underlings.
Partly, because a lot of things just need time to achieve, even the quick ones.
Partly, even for quick achievables, you only have so much bandwidth as administration, so you won't have a large and nice list to point to for a first week or two.
More than 3 months, and the honeymoon periods discussed above are over
More than 3 months, and attention span wavers.