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The Constitution says in Article 2, Section 1: President and Vice President, Clause 2: Method of choosing electors:

Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.

The Twelfth Amendment to the United States Constitution provides the procedure for electing the President and Vice President. It replaced the procedure provided in Article II, Section 1, Clause 3, by which the Electoral College originally functioned. Key text of Amendment XII below...

The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and all persons voted for as Vice-President and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate;
The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted;
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

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"While the Twelfth Amendment did not change the composition of the Electoral College or the duties of the electors, it did change the process whereby a President and a Vice President are elected. The new electoral process was first used for the 1804 election. Each presidential election since has been conducted under the terms of the Twelfth Amendment.
The Twelfth Amendment stipulates that each elector must cast distinct votes for President and Vice President, instead of two votes for President. Additionally, electors may not vote for presidential and vice-presidential candidates who both reside in the elector's state—at least one of them must be an inhabitant of another state."

If a state legislature decides to appoint its electors based on the national popular vote, how exactly does that run afoul of the Constitution? It is completely within the state's purview to decide how to appoint its electors.