Most explanations miss a very simple point. Mercator projection becomes simple if we use complex numbers. It is the stereographic projection which sends the North pole to $\infty$ and the South pole to $0$, followed by the complex logarithm. All formulas immediately follow from this observation.
For the details and historical discussion see Osserman's paper Mathematical Mapping from Mercator to the Millennium,
http://www.msri.org/people/staff/osserman/papers/ROssermanPart_V.pdf