Each state in the United States sends one or more "Representatives" to the United States House of Representatives. The collection of such "Representatives" from a single state is that state's "delegation".
According to the Twelfth Amendment to the U.S. Constitution, if no single candidate for President receives "a majority of the whole number of Electors appointed" to the Electoral College, then the House of Representatives chooses the new President.
But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice.
The article is pointing out that if each state's delegation to the House of Representatives were to vote for President along party lines, it is likely that a majority of the states would vote for a Republican. There are enough Republicans in other states (such as California) that the Republicans could achieve the required quorum.