I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta function that motivated him to make a connection with the holomorphic projection of weight 1/2 harmonic Maass forms? Any reference or help will be highly appreciated.
In addition to Zagier's excellent Bourbaki seminar I would also recommend some notes by Ken Ono that include both a summary of the history of mock theta functions and mock modular forms and a survey of applications. They are available at http://swc.math.arizona.edu/aws/2013/2013OnoNotes.pdf.
You might also want to take a look at Brunier and Funke, "On Two Geometric Theta Lifts," Duke Math. Journal 125 (2004).
Zwegers is on MO but I don't think he is a regular participant. Perhaps someone who knows him will see this and ask him to describe his motivation to you directly.
There is a Zagier's Bourbaki talk "Ramanujan's mock theta functions and their applications" people.mpim-bonn.mpg.de/zagier/files/aster/326/fulltext.pdf . In the article, you can find a comparison between pre-existing quasimodular forms of Kaneko-Zagier and mock modular forms.