That's right, the points are just like True Strike in 5E.
People like being rewarded bonus points. Behold below and see the evidence of my claims! And then see my own method of serving this base, vulgar, hubris-laden need.
Your DM can choose to give you inspiration for a variety of reasons. Typically, DMs award it when you play out your personality traits, give in to the drawbacks presented by a flaw or bond, and otherwise portray your character in a compelling way. Your DM will tell you how you can earn inspiration in the game.
-D&D 5E PHB, pg. 125
A player who enjoys playing a role well may sometimes make decisions that fit his or her character but don’t necessarily lead to the most favorable outcome for that character. … Such roleplaying should be rewarded, since it enhances the game. (If it doesn’t enhance the game, don’t give an award.) XP awards for roleplaying are purely ad hoc. That is, no system exists for assigning Challenge Ratings to bits of roleplaying. The awards should be just large enough for the player to notice them, probably no more than 50 XP per character level per adventure.
-D&D 3E DMG, pgs. 40-41
Mutants & Masterminds reflects this kind of story structure through the awarding of hero points. The heroes gain additional hero points as an adventure progresses. When the going gets tough, the heroes get tougher, because they get hero points to help them overcome future challenges. Heroes get hero points from setbacks, complications, acts of heroism, roleplaying, stunts, and instances of Gamemaster fiat.
-Mutants & Masterminds 2E core rulebook, pg. 122
At the end of each play session, the GM should award bonus character points for good play. “Good play” is anything that advances the heroes’ mission or shows good roleplaying – preferably both. But roleplaying trumps mission success!
-GURPS Lite for Fourth Edition, pg. 17
Pendragon is a social game; Glory represents the respect of one’s peers, not something palpable like gold. Thus, to a great extent, Glory is dependent on the reaction of the Gamemaster and the other players to your roleplaying. This section offers guidelines, but not absolute rules, for earning Glory. If you impress everyone with your character’s actions, chances are that he receives some Glory. … Some reasons to award extra Glory might include the following: … Made the Gamemaster laugh uproariously: Add 10.
Pendragon 5E core rulebook, pgs. 101-102
Each character starts each session with a pool of action dice, as shown on Table 1.4: Career Level (see page 27). Additional action dice may be gained from various character options, and as GM rewards for playing in character, taking chances (especially heroic ones), furthering the story, solving problems, entertaining the group, and otherwise improving everyone’s experience (for more information about action die rewards, see page 365)
-Fantasy Craft core rulebook, pg. 62
You have seen this sort of thing before. And I don't think there's anything wrong with it. I'm not saying that any of the above methods or others you have seen out there are bad.
I'm just saying that I don't care.
Here's what I do: do you remember the show Whose Line is it Anyway? I think it started in the UK, then it moved to the US with Drew Carey in the late 90's and early 2000's, and now it's hosted by Aisha Tyler. And every episode would open with Drew saying, "Welcome to Whose Line is it Anyway, the show where everything's made up and the points don't matter." Then, after each dumb game they played, he awarded everyone an arbitrary number of points. Sometimes in the hundreds. Sometimes in the thousands. Sometimes in the millions. Sometimes, just, like, a handful. It truly was random and didn't make sense. The only prevailing logic was that he seemed to give points to people who did something really great and memorable during that game.
As a Dungeon Master, this is literally what I do. I am just Lich Drew Carey.
It's surprisingly tricky at first to throw out "random" numbers. You really have to train yourself to speak without thinking and not worry about what you're saying. One time I gave out ten thousand points for something and apparently it shocked everyone present. I wouldn't know if that makes sense though because, and I cannot stress this enough, I do not think about or pay attention to the actual points coming out of my mouth. I guess ten thousand was relatively very high for me, but guess what: the points don't matter.
My players like to write down how many points they get and keep a running total. Then, at the end of the session, we find out who had the most overall. It's a fun moment that really helps with the wrap-up. And it doesn't matter.
The points don't do anything. They don't give you a bonus or let you re-roll or give you a hint. They aren't even given for roleplaying, like most of those examples I gave above talk about. I just give it for dumb shit that I like.
And despite this, I watch as my players scramble among themselves for a desperate attempt to appease me and gain these precious, meaningless points.