The story I have put together from public statements, is that Lila, in consultation with at least some of the board, and some part of the WMF staff, did some serious planning toward building a Big Search Engine thing, that they did see as competing with Google, but only for people looking for certain kinds of content - namely knowledge. They apparently kicked around a $32M budget for this to be spent over several years. This was the "Knowledge Engine" in its grandest form. They wanted to connect up with many sources of noncommercial information both within WM properties like WikiVoyage, Commons etc along with external sources (e.g census data) via Wikidata, and present results not as some list of webpages like Google presents, but as a serious Knowledge Graph sort of thing if there was no WP article (a kind of mini-article-on-the-fly); it is not clear to me now, if they planned to eventually present the article-on-the-fly instead of a relevant WP article. But you can see how this would be powerful and cool. The page for this was meant to be wikipedia.org. If you have ever looked at that page (I never do), you see it has a search box right in the middle. That is where they wanted people to come to find "knowledge", and keep coming. And they wanted to do things like get that search box on kindle and other platforms, to draw people here and keep them here. (NB, there were updates to the portal just today March 11 per this)
They wanted to do that in order to get eyeballs coming directly here and staying here, because eyeballs here = donations; some within the WMF views the WM/WP sites as literally helpless to compete for eyeballs in the face of the Knowledge Graph by Google and other repurposers. However, a thing that a tech organization can do within the mission, is create a search engine for knowledge that leads you to content putatively unaffected by commercial interests (I say putative because i have seen nothing about managing promotional editing, and "noncommercial" was a big part of the pitch). So this was being done 1) to sustain the organization in the face of what Lila called an "existential challenge"; 2) to remain "relevant" as a tech-based organization; and 3) it does fit the mission. I am not sure if the folks at WMF who were planning this, also planned to get revenue from the KE directly in some way (would they try to get Amazon to pay to put the wikipedia.org link on Kindle? I am not sure yet)
The first phase of that was always improving intra-WM/WP search, because it sucks and it is one thing that leads people to pop in and then pop out to Google again when they can't find things with our search. Discovery is working on improving our internal search capabilities now.
Apparently the KE effort was siloed within WMF and some of the discontent over Lila among staff had to do that weirdness resulting from that. I understand that it was also siloed within the Board. As I understand it, some of James' efforts to understand all this and its effects on staff were upsetting to the other board members, and it was those efforts (I think they would say, it was not so much why he was doing it, but what he was doing) that led to his dismissal. From James' perspective, his efforts to get transparency around all this stuff, both in the community and among WMF staff, were a big part of why he was dismissed. And also, his perspective that the community should get input into whether so much money should go into this big KE thing, or go into tools to help us create more high quality content and get that content out there (e.g. investment in mobile viewing and editing).
There is a sideshow about Damon, who was the head of engineering on the WMF staff, hired by Lila, who was apparently internally shopping around a full blown search engine (not just noncommercial content, but kind of a Mozilla for search generally) in a real cloak and dagger way, but I think that didn't get far and is not the heart of this thing.
That's the big picture as I understand it.