I don’t understand why Apple wouldn’t break into this phone without telling the government how they did it.
That's not what the Court Ordered. The Court order required Apple defeat its own lock-out system and give that software to defeat that system to the FBI.
In addition, if Apple had done that, it would be considered a Forensic Lab, and would be required to backup exactly what it did in court, giving the defense counsel of anyone arrested using the evidence gleaned from such a search, the code used to investigate exactly what that code was capable of doing such as planting evidence, etc. It would be impossible to keep it secret. This has actually happened in the past. . . a secret method was used to decrypt data on a Blackberry to convict a child molester, under the agreement the software used to unlock the data would be destroyed. The judge ordered it released to the defense under seal. The Defense's IT guy thought it was cool and kept a copy and shared with a few buddies, the defense lawyer kept a copy, and after the trial, the press sued to have it released under the "Public's Right to Know" and the judge agreed and ordered copies released to the press. . . despite the iron clad agreement it was going to be a one-time exception and it was guaranteed to be destroyed after use. OOPS!
Apple DIDN’T do it, a third party did.