How does John The Ripper work when trying to crack passphrase of a private ssh key? As in what steps are involved when it tries to do so? What's the role of ssh2john in the whole process?
ssh2john is a utility to convert the key-file into a txt-format that would be suitable for JtR to crack by comparing hashes. There are also other utilities available e.g. password protected zip-files, keepass DBs etc.
The steps really depends on which mode you use to crack, you could use e.g. word-lists or increment mode. see https://www.openwall.com/john/doc/MODES.shtml