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