Here's an attempt at an explanation/translation:

HTP ("Hack The Planet") is a group that likes to break into things. Another (unnamed) group of people impersonated a third group of people ("ac1db1tch3z") and tried to cause trouble for HTP.

The impersonators located HTP by examining one of HTP's botnets (a collection of compromised computers that are used to launch things like denial of service attacks). Botnets have to receive instructions (e.g., targets to attack) from somewhere, so it's likely that the impersonators followed the path taken by commands to the botnet, and found the network(s) that HTP uses to organize themselves.

HTP realized this, and wanted to get back at the impersonators. They found out that the impersonators used an IRC channel (chat room) hosted on a network called SwiftIRC. If HTP could break into SwiftIRC (which is hosted on Linode), they could cause all sorts of trouble for the impersonators. So HTP decided to break into Linode, so they could break into SwiftIRC, so they could break into the group of impersonators.

To break into Linode, HTP broke into their domain name registar (name.com). They planned to secretly take control of linode.com, and replace it with a version of linode.com would look and feel and work correctly, but had one additional feature -- it would collect the login information that people typed in. HTP probably hoped to gain the login for SwiftIRC directly, or collect the logins for Linode admins and obtain SwiftIRC's login from there.

But, before they enacted the domain takeover (a maneuver that would likely be somewhat difficult to employ without being noticed), an HTP member discovered a new vulnerability in ColdFusion, the server software used by Linode. The ability to discover a new exploit on demand implies a high level of skill within the group. Using this exploit, HTP obtained direct access to Linode. They proceeded to gain access to SwiftIRC, as well as other sites hosted on Linode, including a well-known security site, nmap.org

The FBI apparently had a mole in HTP, and they alerted Linode that HTP had access to nmap.org. This posed a bit of a problem for HTP: if it became public knowledge that they had obtained access to Linode, then perhaps they wouldn't have time to go after the impersonators using their newfound access to SwiftIRC. So, HTP tried to strong-arm Linode into staying quiet until May 1st. HTP had obtained the customer information and credit cards of all the Linode customers. HTP threatened to widely publish all this sensitive information if Linode didn't stay quiet. If Linode complied, then HTP would just delete all the info.

Linode, though, was forced by the FBI to announce that they'd been broken into. HTP told Linode to just publicly acknowledge that HTP was the group that broke into Linode, and they'd delete the sensitive info. Linode did so (https://blog.linode.com/2013/04/16/security-incident-update/).

HTP conducted an internal investigation to determine which group member(s) were working with the FBI. HTP broke into the mole's computer and turned on their webcam, and saw an FBI employee looking over the shoulder of the mole. They kicked the mole out of the group, so the FBI doesn't have access to HTP anymore.

(Remember, this is the story according to HTP.)