Bloom Gist has been searching and gathering answers to this widely asked question and most of the answers provided by people has failed to be widely accepted since people tend to see precise answers on how they profit by it rather than general answers like, they do it by collecting users data.
So, Bloom Gist has came up with an answer which is considered to be one of the key ways by which these web giants earn money through their web browsers and how they (like Google and Mozilla) might lose huge some of it's revenues if their major competitors make a slide move off their current base.
Lets start with Google.
Google is a company based entirely on Web. Almost all of Google's product - Gmail, Google Search, Google Maps, Google Docs, Google Drive, Google+, Hangouts, Youtube are based out web. Also, most of its profit comes because of web (mostly ads).
Since all the products are web based, they need to be run on a browser. Before Chrome, browser market was dominated by IE and Firefox. As Jeff correctly pointed out, its not good if your success depends so much on an application (browser) which is not controlled by you. However, I don't think Google feared of some anti-competitive changes in IE as much as it feared of the (lack of)progress of IE. By progress, I mean supporting new, open web based technologies (html5, native video playing capabilities, etc.).
Though Mozilla claimed that its dedicated to supporting new technologies, the progress wasn't satisfactory. IE on the other hand never bothered much about it anyway. But support for these new technologies was crucial for Google to enhance its user experience and also to allow more rich set of applications (e.g. Google Docs, Google Sheets, etc) to be run on browser.
Also browsers were getting more and more bloated and slow, resulting in bad user experience which in turn force people to use internet less and less which in turn was not good for Google. So, in a way, browsers were acting as a bottleneck for expansion of Google's business (and their ultimate aim of building a StarTrek Computer) and Google could do nothing about it.
This was before Google decided to take things into their own hands and created a fast and beautiful browser - Chrome. In the beginning it was all about speed because that was what Google was mainly concerned about. Then it started adding support for latest web technologies in Chrome with a breathtaking pace. Speed combined with some really innovative features (Omnibar, The combined wrench menu, the clutter free design, the separate-tabs-in-separate-process design) made Chrome rise through the ranks and it soon became the number one browser in market. With this, Google gained control on one of the most important missing piece of their business model.
The popularity of Chrome also forced other browsers to improve quickly. Mozilla changed its release cycle to be of similar period as that of Chrome (Remember how we had only Mozilla 4.0 from 2002 to 2010 and how we are running Mozilla 29 right now). IE also improved quickly as compared to its historical pace. This was good for Google as it improved user experience on the web which in turn benefited mostly Google and its web products.
All the above points to non-tangible earnings for Google.
Now about the tangible earnings (actual money):-
Chrome also allowed Google to collect more and more data about users (the omni bar is brilliant but also sends all the keyinput to Google even if the input is not a search query but a website that user directly want to go to). This allows Google to serve better targeted ads and in-turn earn more profits.
Lets move on to Microsoft and IE
Few things are similar to Google and Chrome -
1. Default search is Bing. So earn money through ads.
2. Learn more about users browsing habits, preferences, etc. Serve better targetted ads - earn more money.
3. Having the most dominant product in market is always beneficial. You can bend lot of rules according to your wish. Promote technologies which are beneficial to you, etc.
Now something different:-
Microsoft has a lot of products / apps / systems designed for enterprises (most of them run on IIS - microsoft's own webserver). Also, IE support something called ActiveX which in my opinion is the ugliest thing I have ever see. Lot of other enterprise focused companies have designed many tools which are served by IIS (and many of them use ActiveX). You could see that in your own company's internal portal - the employee management system, the workflow management system, the web encryption system, the bloated ActiveX's and what not. Most of these products do not run properly, if at all, in other browsers. For Microsoft to continue to earn money from enterprises, it is necessary that they continue to maintain a browser which can run all these ugly products. So IE.
Now Mozilla:-
They mostly earn money through donations and from Google. Google pays them money to keep Google search as the default search engine (Just like it pays money to Apple to keep Google as default search engine on iOS). Google in-turn earns money by serving ads to people using Google search on Firefox.
