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. 

Indeed, Mercator's projection is characterized by two properties a) it is conformal, and
b) meridians and parallels are mapped to perpendicular lines.
Stereographic projection maps meridians to lines through the origin, and parallels
to circles centered at 0. Complex logarithm maps them to horizontal and vertical lines respectively.

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