As the old saying goes, "Give 'em enough rope, and they'll likely hang themselves." Sometimes this is the best approach when dealing with blocked users. If they are pleading to be unblocked and swearing up and down that they understand and won't repeat whatever it was that got them blocked, rather than arguing the finer points of the original block or demanding further explanation, it's usually better[note 1] to just unblock them and make it clear that this is their last chance. If they mean what they say, then unblocking will be the right thing to do, and if they don't, they'll be blocked again soon enough.
There's plenty of rope to go around; don't be afraid to hand it out.
Sometimes there is a discussion about whether an editor should be blocked or warned. When in doubt, a warning is probably sufficient. If a block is needed, the editor will soon prove it.
Example scenarios
Example #1: The music fan
User:Ididntdoit starts adding excessive external links and original research to multiple articles on their favorite musicians, receiving numerous warnings in the process.
Admin User:SpamHunter007 blocks them when they continue after a final warning.
Ididntdoit asks almost instantly to be unblocked. In their request they state that they now understand what they did wrong, and promise not to do it again, adding that they are really, really sorry and didn't know this was against the rules.
User:Bigbro has a history of making infrequent edits. Not particularly helpful, but not really doing any harm.
Late one Saturday night Bigbro begins adding the same rhyming couplet containing a very nasty racial slur to numerous, seemingly unrelated articles.
Two days into his three-month long block for vandalizing, he uses the classic argument that his little brother did it while he was away, and now little brother is in big trouble.
Most likely reactions
The admin reviewing these requests has several choices before them:
Decline because they believe the editors to be lying.
Prolonged discussion until the user and the reviewing admin have gone over nearly every edit the user has made and analyzed them point-by-point.
Unblock and give them the chance to prove that they do understand and can refrain from repeating this same behavior, or conversely a chance to prove they do not understand or do not care and were just hoping for a sympathetic admin to unblock them. Little do they know they have "hanged themselves" and will be rapidly re-blocked, this time probably indefinitely.
This is not to say that this is always the better solution. Sometimes those prolonged unblock discussions produce real results in educating the blocked user about why they were blocked and helping them to edit productively in the future. But as a simple litmus test of a user's sincerity and willingness to edit cooperatively, it can be very effective. Sometimes even after a prolonged discussion it's hard to tell if the user understands the problem and/or is willing to stop editing disruptively, and the only way to find out for sure is to give them the opportunity. Another option in the case of vandal accounts is to decline the unblock request but add {{2nd chance}} which basically asks them to prove it before being unblocked.
When not to use
If a user has already been blocked numerous times for the same behavior, they've already gotten all the rope they need; the hangman is just asleep at the lever.
If the user was justifiably blocked but is not giving any indication that they even feel they did anything wrong,
The intention of this essay is to assist administrators when reviewing unblock requests, and discussions between admins about unblock requests is where it has often been cited. It could be viewed as uncivil to cite it in a direct discussion with the blocked user.
Notes
^In circumstances that are not serious (vandalism, disruptive editing, edit warring) versus serious violations (such as egregious, blatant, or libelous BLP violations, sockpuppetry, or long-term abuse)
