According to the Washington Post, sources familiar with the matter, have stated that the initially suspected collaboration with Cellebrite is not how the data from the encrypted iPhone was recovered. Instead an unknown security vulnerability was used (purchased) from "professional hackers" to prevent the phone from erasing its data and slowing down the passcode check. This would mean that the FBI probably brute-forced the passcode once those iPhone security measures had been deactivated.
@Gandalf brought to our attention that the DOJ has now confirmed they successfully accessed the iPhone 5C and can indeed access any iPhone 5C running iOS 9 this way: http://money.cnn.com/2016/03/28/news/companies/fbi-apple-iphone-case-cracked/index.html
There is a good article about the current known facts here:
The official declined to provide any details, such as what was on the
phone, the identity of the third party or how the method worked. The
official also declined to say whether the U.S. will give Apple details
about the hacking method.
According to other news reports the FBI decided to go with a company that stated it could unlock the phone for them. Something that had been suggested to be possible for a while. They did not specify which company / third party is going to do this, but several articles suggest it's an Israeli Company, named "Cellebrite":
Original Report: http://www.ynetnews.com/articles/0,7340,L-4782246,00.html
The speculation is suggesting NAND mirroring to be used to gain access to the device memory and copy everything to another device, but at this point the exact how-to is not publicly known.
This article also found a recent contract reference between that company and the FBI: http://www.macrumors.com/2016/03/23/fbi-israeli-firm-cellebrite-to-unlock-iphone/
They also speculate about a microchip attack and brute-forcing the passcode once the data has been extracted from the device, but the company is likely going to remain very quiet about the exact "how-to" and so will the FBI for obvious reasons.
The EFF has published a statement that suggests any used security vulnerability might be reported to Apple, if the FBI was to follow the guidelines that the EFF had previously acquired through a freedom-of-information lawsuit: https://www.eff.org/deeplinks/2016/03/fbi-breaks-iphone-and-we-have-some-questions