According to Section 8(a)(1) of the NLRA, employees can't generally be disciplined for exercising the rights provided under Section 7 of the NLRA. These rights are commonly understood to be about unionizing (ie, you can't be fired for trying reasonably to organize a union), but are actually broader: the NLRA protects an employee right to almost arbitrary "concerted action" to improve working conditions.
Damore was terminated by Google for authoring his anti-diversity memo. During the time he was authoring and distributing internal copies of the memo, he worked with (apparently) a bunch of other engineers at Google that shared many of his viewpoints (the memo covers a lot of ground). He was terminated after the memo, an artifact of his concerted effort to change aspects of how Google was managed, was published. He and his lawyer mounted an 8(a)(1) complaint.
As a starting point --- people with real-world experience or understanding of the NLRA should correct me where I'm wrong --- NLRA complaints get filed with the National Labor Relations Board. The NLRB adjudicates claims internally. If a complaint is found valid, the NLRB will try to convince the employer to settle with the employee; if that goes nowhere, the NLRB will itself effectively sue the company. If the complaint is found invalid, the NLRB will inform the complainant that NLRB is done with the case.
Most of these cases (there are tens of thousands annually as you'd expect) are handled by grunts in DC. Novel or high-profile cases are escalated to a special department in the NLRB called the Advice Division. This particular case was not only escalated but apparently handled personally by the director of the division, an NLRB lawyer who'd been with the board since 1988.
The NLRB turned down Damore's complaint. I think he could theoretically still sue in civil court? I don't think that's commonly done? Either way, his complaint was denied.
The basis for the complaint is super simple and kind of obvious in retrospect: employers in the US are required by federal and usually state law to avoid discrimination against protected classes. To the extent that Damore's memo was about organizing against discrimination of conservative viewpoints --- a phenomenon that is almost certainly real in SFBA tech! --- it was protected. But to the extent that it attempted to organize around changes to Google management that might (might) themselves violate EEO laws (for instance, any kind of official recognition that men are better suited to software development at Google than women), they were not. You can't use the NLRA to organize in opposition to federal employment law. Wa-waa.
As it turns out, this was apparently super-apparent to Google legal and Google HR, who fired Damore precisely by the book, exclusively for promoting stereotypes about women and advocating for the inclusion of those stereotypes into Google's management processes.
I think an important thing to consider --- I'm no lawyer and am probably wrong about lots of this stuff --- is that if the memo had been exclusively about how SFBA tech discriminates against conservatives and could in a number of ways be made more accommodating to them, Damore would had been protected from retaliation. The Advice Memo says as much!
So for someone like me, who believes very strongly in both the importance of employee organizing rights (I think tech should organize into professional associations; "unions lite") and who believes strongly in the absolute innate equivalence in aptitude for our profession between men and women, this is the best possible outcome. There's a lot to take heart in here; we do in fact have the right to organize that we've been saying we have.