What tactics did the Secret App use to grow it's install base so quickly?

To put it simply: their VCs.

Take a look at who funded them:

Seed (12/2013) $1.43M
Google Ventures
Initialized Capital
SV Angel
Brett Slatkin
Harry Cheung
Fuel Capital
Index Ventures
Kleiner Perkins Caufield & Byers
S-Cubed Capital

Series A (3/2014) $8.6M
Kleiner Perkins Caufield & Byers
Garry Tan
Alexis Ohanian
MG Siegler
Bing Gordon
Megan Quinn
Chris Howard
Brad Silverberg
Vivi Nevo
SV Angel
Ashton Kutcher
David Sacks
Bill Lee
Pete Cashmore
Joe Montana
Rob Wiesenthal
Google Ventures

(from http://www.crunchbase.com/company/secret )

Many from this list are using the app, talking about the app (very often), and creating the air of inevitability. Their social networks are extremely influential. There were apps similar to Secret before, but none with such A List supporters behind it.

Most companies don't disclose their launch strategies, for obvious reasons, so the following is opinion/conjecture based on my own experience:

Part of Secret's success was releasing a novel app that had some contextually relevant cultural value at the time of it's release--the idea of anonymity in the digital social space. Mix that with getting the app into the hands of target-relevant influencers, a powerful PR campaign and a little bit of luck that it got picked up by the tech media.

Though in the end, due to a quick growth on an idea that had not been really properly vetted and tested secret turned into something ugly:

http://www.dailymail.co.uk/sciencetech/article-3064077/The-secret-s-Anonymous-app-shuts-following-legal-battles-claims-encouraged-bullying.html

The key to the success of The Secret apps is that they provide a user experience that is consistent and faithful to the core concept and themes of The Secret series of books in a practical and accessible manner. The mobile apps have allowed developers to use pre-existing creative content within a new platform to generate solid and dependable revenues streams with the key advantage of immediate financial reporting and direct response to our social media marketing strategies.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath