I cannot in good conscience obey an order that I believe violates the sacred oath I took in the presence of my family, my flag and my faith to support and defend the Constitution of the United States.

From the top. There are five names you should know:
Eddie Gallagher, a Navy SEAL, accused of war crimes,
Several spots above him is Rear Adm Collin Green, who oversees the SEALs and has made solving order and discipline problems a priority
A few spots above him sits Richard Spencer, who is was Secretary of the Navy,
Above him is Mark Esper, who is the Secretary of Defense,
who reports to Donald Trump, the President of the United States and Commander in Chief
​

• Gallagher was accused of various war crimes by several of his own men, was tried and was exonerated on all but one count.
• Recently, Donald Trump commuted Gallagher's sentence.
• Earlier this week Spencer reported that the Navy was reviewing the Gallagher case, and this review that might lead Gallagher to be ejected from the Seals. Historically, these reviews have almost never ended favorably for the individual in question.
• Trump became aware of this and tweeted that Gallagher would not face these consequences.
• Spencer stated that "a tweet is not an order", and that the review would continue.
• Various media report that Spencer and Green - both of whom are presumably interested in protecting the institutional integrity of the Navy - have threatened to resign if the review is halted. This would have been a significant, historic rebuke. Spencer publicly denies this. Green has not commented publicly, but other officials deny it on his behalf.
• Spencer is fired by Esper. Esper's justification: Spencer went behind his back and proposed a "sham review" to the WH, where Gallagher would ultimately be exonerated. This story does not match up with Spencer's earlier public statements supporting the review.
• Esper states that the review will be cancelled, and Gallagher will retire as a SEAL. This is the outcome he claims that Spencer was pursuing through illicit means, and is the outcome that Gallagher and the President wanted. This is not the outcome that Spencer seemed to want in his public statements.
• Esper's justification of Spencer's firing differs dramatically from Trump's (given in another series of tweets).
• More importantly, Esper's story differs dramatically from Spencer's termination letter. His letter is damning, and more-or-less accuses President Trump of giving an illegal order:

Unfortunately it has become apparent that in the respect, I no longer share the same understanding with the Commander in Chief who appointed me, in regards to the key principle of good order and discipline. I cannot in good conscience obey an order that I believe violates the sacred oath I took in the presence of my family, my flag and my faith to support and defend the Constitution of the United States.

It's not clear what happened, but if you believe Spencer it sure sounds like Trump followed up his tweet with a formal order - perhaps to stop the review, or to force a "sham trial" - and Spencer refused, hence his termination.