The President's constitutional position is defined in article 2.
Section 1 states "He shall hold his office during the term of four years"
It then describes the process of electors casting votes and as amended by the 12 amendment states "the person having the greatest number of votes for President, shall be the President; if such number be a majority of the whole number of electors appointed."
And finally, this paragraph is further amended by the 20th amendment which states "The terms of the President and Vice President shall end at noon on the 20th day of January"
So there is the constitutional basis for a change in President. The length of one term is 4 years. The term ends on 20th January. The new president is chosen by electors.
A concession speech is important because of another part of the Constitution. The Presidential oath. The President swears to "preserve, protect and defend the Constitution of the United States." This means that the President swears a solemn oath to (among other things) leave the office of President at the end of their term unless re-elected. In the past, presidents have fulfilled their oath in part by giving a concession speech saying "The other guy won, and I will recognise that". It isn't required, as the President can "preserve protect and defend" without a concession speech. However, it is unprecedented in modern times for a defeated President to refuse to do so, for it risks conflict and bloodshed.