The wikipedia article has most of the salient points:
In September 1788, after the necessary nine states had ratified the Constitution, the Congress of the Confederation set March 4, 1789, as the date "for commencing proceedings" of the newly reorganized government. Despite the fact that the new Congress and presidential administration did not begin operation until April, March 4 was deemed to be the beginning of the newly elected officials' terms of office, and thus of the terms of their successors.
The Constitution did not specify a date for federal elections, but by the time of the second presidential election in 1792, Congress had passed a law requiring presidential electors to be chosen during November or early December. By 1845, this was narrowed to a single day, in early November. Congressional elections were generally held on the same day.
So, there are 4 months between congressional/federal elections and the new administration taking office. To compound the lame duck issue this presents:
Article I, Section 4, Clause 2 mandated a Congressional meeting every December, after the election but before Congressional terms of office had expired, a lame duck session was required by the Constitution in even-numbered years; the next session wasn't required until the next December, meaning that new members of Congress might not begin their work until more than a year after they had been elected.
This was less of a problem in the 19th century - basically, congress was held one long session (a few months) starting in December of even years, and a shorter lame duck session in odd years following the congressional elections. This was basically due to the travel time, particularly for representatives in the west, to get to Washington.
However, the lame duck period did present problems at times:
most notably in 1861 and 1933, after the elections of Abraham Lincoln and Franklin D. Roosevelt, respectively, plus the newly elected Senators and Representatives. Under the Constitution at the time, these presidents had to wait four months before they and the incoming Congresses could deal with the secession of Southern states and the Great Depression respectively.
As transportation improved in the 20th century, and congressional sessions started being held almost continuously, the 20th amendment came about to shorten the lame duck period following the presidential election.
Ironically, it seems the immediate effect of ratification was to massively delay the first session of 73rd Congress - causing Roosevelt to convene a special 100-day session starting in March 1933 rather than wait until January 1934 as the amendment specified.