First, a reminder of the purpose of those writings. Madison and his co-authors, Alexander Hamilton and John Jay, were trying to persuade New Yorkers to ratify the Constitution that had been drafted in Philadelphia in 1787. So, of course, Madison wants to make the Senate sound like the neatest thing since sliced bread (sliced bread didn't exist yet, but you know what I mean).
Madison notes the stricter qualifications for senators, as compared to House members. There is a higher minimum age, and a longer minimum period of citizenship. "The nature of the senatorial trust ... requires ... that a senator should have reached a period of life most likely to supply" the necessary "information and stability of character". He evidently was referring to the powers given exclusively to the Senate, such as the requirements that treaties and presidential appointments get Senate consent.
The Constitution requires that a person must have been a U.S. citizen for at least nine years, to be eligible to become a senator. Also, the Senate was destined to take the lead on foreign policy issues, with its powers related to the appointment of ambassadors and the ratification of treaties. Madison takes both of those things into account, in noting that the citizenship requirement prevents someone from emigrating to the U.S., and then immediately playing such a major role in implementing American foreign policy.
That was more of an issue in the 18th century than it is today. Royalty had more political power back then, and they were more likely to move among countries. The founding fathers were presumably mainly thinking about their archenemy, the British King George III, whose grandfather had moved from Hannover, part of what is now Germany, to take the British throne in 1714.
I don't recall any controversies in modern times regarding the immigrant status of U.S. senators. Two secretaries of state during the last half-century were immigrants: Henry Kissinger and Madeleine Albright. Kissinger's status drew more attention in that he retained a strong German accent. At one point, the Nixon Admnistration in which he served, wanted the TV networks to show silent video of Kissinger, so as to play down his accent. That idea was short-lived, and there was little, if any, serious objection to having an immigrant in that job.
Madison moves on to discuss the method of electing senators, which has changed in the meantime. Originally, senators were elected by state legislatures. He attributes to that method the "advantage of favoring a select appointment". This is yet another example of the founders' advocacy of decision-making by political elites, rather than the great unwashed.
I find more interesting his second point, about "giving to the State governments such an agency in the formation of the federal government as must secure the authority of the former, and may form a convenient link between the two systems". As I've noted in earlier posts, the states were the dominant players in American politics at that time, and the fledgling federal government needed to secure their support.
Effective with the 1914 election, a constitutional amendment took the power of electing senators away from legislators, and gave it to the general electorate of each state.
Madison's next subject is the states' equal representation in the Senate, in contrast to the apportionment of House seats according to population. That was the result of a pragmatic compromise, but Madison does a fair job of fitting it into a theoretical framework.
An international organization will typically give equal representation to member states, regardless of population, GDP, or whatever. Many years later, the United Nations General Assembly was established with such a structure. On the other hand, a centralized government would allocate representation purely according to population. Madison sees the federal government under the Constitution as falling between those two extremes, and therefore the different structures of the Senate and House fit with that middle-ground approach.
He goes on to make what in these times we would call a libertarian argument. The requirement that legislation pass both houses of Congress requires approval by both a majority of the population, and a majority of the states. This he sees as a bulwark against an "excess of law-making". While the bicameral Congress probably performs that function better than a unicameral legislature would, many of us consider Congress's law-making, especially during the last 75 years, to be excessive.
Madison then argues that creating a second house of Congress decreases the likelihood of yielding "to the impulse of sudden and violent passions". And giving senators longer terms that do not all expire simultaneously helps ensure that the body will have "due acquaintance with the objects and principles of legislation". Too much turnover among the membership means those in Congress are less likely to know what they're doing.
From the perspective of today, that concern on Madison's part seems unfounded. The accumulation of perquisites of office, most notably congressional pensions, has created a system where many legislators want to stay in Congress forever.
Of the 10 longest-serving senators in history, four are current senators, including #1, Robert Byrd, Democrat of West Virginia. He has been in the Senate longer than Barack Obama has been alive. The six retired or deceased senators on that list have all left the Senate during the last forty years.