There are two separate kinds of legal protection at issue here, copyright and patent. The only thing that's close to covering ideas is patent law, not copyright law. (I'm assuming US law here, since that's where Facebook and I both are, but most of this is generally true I think.)
Copyright covers a creative expression, and is automatic. If I write a story, paint a picture, compose a song, or code a program, copyright secures my ability to commercialize that creative work as I see fit. If someone else writes a similar story, paints a similar picture, etc. but does not use my words or notes or lines of code, they aren't infringing my "copy right", because they're not copying my work.
Patents cover inventions, and are very non-automatic and best acquired with lawyers. Their term is also much shorter (about a decade, instead of about a century). US law currently lets you patent an algorithm or an arrangement of computer systems, under the artifice that you're actually patenting the implementation of that algorithm on any physical computers. Patents are issued for the invention, not for a specific implementation, and any implementation that works the same way infringes the patent.
(Trademarks, for reference, cover names used in advertising, i.e., "trade marks". They're somewhat automatic, they only cover names, and they're not very often relevant to F/OSS licenses.)
Traditionally, F/OSS licenses have been primarily concerned with copyright, partly because several of them were written when software copyrights were well-acknowledged and enforced, but when software patents were rare or nonexistent. This leaves you in the unfortunate situation where e.g. Facebook can write some code (automatically gaining a copyright), get a patent on the idea, open-source the code, and then sue you for patent infringement despite your copyright license. To avoid that, Facebook has an explicit patent grant in addition to the copyright license. (Although, as other commenters have pointed out, F/OSS licenses that don't specifically restrict themselves to copyright can potentially be read as a implicit patent grant if you stare at it hard enough.)
At no point in this does Facebook claim ownership of your work. Certainly your idea isn't legally owned by anyone, either you or by Facebook, if you haven't gotten a patent on it. Facebook's patents remain Facebook's patents, with or without this clause.
Because patents cover any implementation of an invention (whether or not the implementor knew about the patent), and because patents are in stupidly dense legalese, it's super common for people to be inadvertently infringing a bunch of patents. The traditional solution for this is that large companies have defensive patent portfolios in a sort of mutually-assured-destruction scenario: if Facebook sues Google over a patent, Google will countersue over all the patents they have. So there's an armistice, at least among big companies. The explicit patent grant weakens that, because now Google can just incorporate parts of React into their React-competitor, and Facebook loses a patent from their defensive portfolio.
So a few clauses like these are normal: you get a patent grant from Facebook, as long as you don't sue Facebook over (different) patent infringement. This maintains the armistice between big companies, and it's also great for the little guy, who wasn't going to sue anyway because they don't have any/many patents (but also doesn't have a defensive portfolio because they don't have any/many patents). These clauses only cover claims of patent infringement, which are the thing that everyone is unintentionally doing en masse, not copyright infringement, which is much harder to do unintentionally, so they're pretty fair.
But the fourth clause is super weird. It means that you can't dismantle any of Facebook's patents by e.g. providing prior art, without losing your patent license. It might even mean you can't lobby for software patent reform. It solely holds up Facebook's patent portfolio and the armistice between big companies. Usually dismantling patents is a good thing for everyone, because it's disarmament, and it's unfortunate this license doesn't extend to people who do that.
(IANAL, TINLA, corrections graciously appreciated.)
If that reading is correct, you can still use React Native to the extent that doing so doesn't violate any of Facebook's patents. They could argue it does and you could argue it doesn't. Until and unless a judge ruled that it did, you could continue using the software.
Edit: Geofft that's a good point. I suppose this really is a poison pill for a sufficiently large company.
For Google's use case, it means that if they ever find themselves in court claiming to have prior art on any of Facebook's patents (let's say Facebook sues them over something G+ does, and Google says it was described in the literature well before either of them), they automatically lose the (patent) rights to all of Facebook's OSS, even if they're legally in the right. So they can't build anything they care about on React.
"That Facebook patent is invalid." or "Software patents are invalid."
It means that you lose the additional license to use any of Facebook's patents that may cover React. These may or may not exist.
Also, they'll likely be patenting whatever they can out of this technology, whether it's valid or not, you'd have to fight any action that Facebook holds against you - in the meantime invalidating everything, allowing Facebook to sue you for violating potentially dozens of patents; most everyone can't afford a lawsuit.
Facebook open sourcing all of these things is purely a power play, and not trying to contribute to the developer community. The power that this is going to allow Facebook to leverage in the future could get really nasty - it's why Google isn't using any of it.
As an example, say Facebook owns patent 1234567 "Method and system for reacting natively". Google owns 7654321 "High-density computers in a data center". Google builds the new Maps app using React Native. This would infringe 1234567, except that React Native comes with a patent grant, so the Maps app is properly licensed.
Then Google sues Facebook over Open Compute Project, claiming infringement of patent 7654321. This has nothing to do with React Native, but Google has just lost its license to 1234567. So Facebook now gets to sue Google over the Maps app. (More likely, Facebook will threaten to sue in order to force a settlement for the other lawsuit.)
And even if we assume that to be true and assume that both parties believe in patents want a robust, well-functioning software patent system, that clause still sucks. If Facebook and I both invent something at roughly the same time and apply for patents, and the Patent Office misreads our applications and grants them both, and I point this out to them and I had priority (thereby invalidating some of Facebook's claims), Facebook gets to terminate all my patent licenses.
> The license… will terminate… for anyone that [claims] infringement of any patent… by any party if such claim arises… from any software, product or service of Facebook.
> The license… will terminate… for anyone that [claims] infringement of any patent… by any party relating to the Software.
> The license… will terminate… for anyone that [claims] that any right in any patent claim of Facebook is invalid or unenforceable.
Wow.