Your source makes it sound as if Breckinridge's candidacy was some fringe outsider challenge for the nomination. In reality, Breckinridge was the INCUMBENT vice president from a moderate border state (Kentucky) and thus his party's heir apparent for the nomination. If anything, Breckinridge was the status quo candidate in 1860.
It is said that Douglas and Breckinridge were the favorite candidates; however, the convention broke up over the slavery plank (with Southerners insisting that no government - federal, state, or local - could outlaw slavery in the territories) before the balloting for the Presidential nomination began. Several authors suggest that the real reason behind the break-up was that the Southerners did not have the strength any longer to enforce their will in the convention and were unwilling to compromise. Likewise, they did not have the voting strength to force a Breckinridge candidacy. When the Democrats reconvened in their traditional convention city of Baltimore, the remaining Southerners again walked out.
The split in the Democrat Party, as much as anything else, ensured Lincoln's election as President.
As it was, Breckinridge was somewhat of a reluctant candidate and eventually garnered only 18% of the popular vote. Douglas, the mainsteam candidate of the Democrats was able to take nearly 30% of the popular vote, but only carried Missouri and split electoral votes from New Jersey. The combined popular vote of Breckinridge and Douglas would have swamped Lincoln, but a careful analysis of the distribution of votes indicates it would have remained a close Lincoln victory in the Electoral College. Even if Douglas had all of Breckinridge's electoral votes, and Bell's too, he still was short (pardon the pun). Of course, there is no way to know how well Breckinridge would have polled in the North instead of Douglas. He may possibly have taken one or two states Lincoln won in close voting, but more likely, he would have done no better than Douglas.
The bottom line is that the breakaway Breckinridge candidacy facilitated Lincoln's election.