I very much like the explanation that Pedro G. Ferreira, a professor of astrophysics, gives in The Perfect Theory: A Century of Geniuses and the Battle over General Relativity.
Chapter 3, “Correct Mathematics, Abominable Physics,” starting on page 28, outlines the cosmological constant. Although only the first six pages are given online, they cover the reasoning behind it and why others quickly realized what the equations really implied.