Similar to Dungeons and Dragons, Hero System characters advance based on acquiring experience points. However, where as in D&D experience slowly accumulates and is used as a determiner for when a particular threshold is reached to trigger a new level, Hero System characters can choose to spend the points as they come in. There is usually a place on the official character sheets to keep track of how much experience has been spent as time goes on so that you can do a cost check further on and tell whether the character has too many or too few points (it does occasionally happen that points get lost in the shuffle).
Knight 8-, Has connections with Major Institutions
Priestess 8-, Has connections with Major Insitutions
(0/15)Petrifying Stare II - Severe Transform: 1d6, Target must be able to see the gorgon(-1/2), Partial Transform (+1/2), Target Heals Back Normally
The extended breathing could be breathing exercises they have picked up in the course of their adventures, possibly at the same time that they picked up the martial arts. Alternately, they might have ended a session with the possibility of having to traverse an underground river and wanted to be sure they could hold their breath for a longer time and at the end of the session that introduced that possibility, spent experience to take a power to make it easier, linking it back to that earlier martial training. This is perfectly allowable and helps simulate a lot of the fiction Hero System is meant to reproduce.
Now we come to places where experience points weren't spent.
In the case of complications, it is possible to spend experience to reduce the value of Complications and make them less troublesome. Possible but I wouldn't recommend it at such a low number of Complications. I usually end up having extra complications beyond the point value I am required to have, such extras provide 0 extra points, but I do find complications are great for giving a character personality. So buying them off is a bit like saying "let's make this character more boring now." However, sometimes a particular complication might just not be appearing in the game or else might no longer make sense and then you and the GM might want to remove or redesign that complication to better fit what is going on.
In this case, with the increased reputation and contacts, the GM likely felt that the "Extreme Reaction" part of the distinct appearance no longer fit. Certainly some people might still be rude or distrustful, but the fact that she is well known around the city as a hero now means people aren't just going to break out the torches and pitchforks when they see her. Talking, they agree to reduce that to a "Major Reaction" described as "distrust" or "invasive curiosity." This left 5 points that needed to be reassigned, the player might have suggested reducing the Cynic down a bit since this increased socialization and the Conversation skill pointed to the character being less bitter and got another 5 points out of that so they could take a "Budding Affection" complication for some NPC or another.
Likewise, it isn't uncommon for the GM in Hero System to award contacts, wealth, and other perks as part of the natural development of the story. These are generally viewed by players and GMs as bonus experience points which have a designated way to be spent. However, the sanctity of points still applies. If the priestess or knight were to die, the character would likely get nothing to replace them. Meanwhile if the tavern-keeper were to die, the gorgon would be able to reassign those points elsewhere.
The last thing here is the "Possible" which represents something that the character plans to put into the character but doesn't yet have the points to do it. I usually have one or two of these while running a character and I often set aside experience points in order to save up for such things. In this case I didn't do that but wanted an example of such anyway. I have noted that the player has spent 0 out of the 15 necessary points to get that power. It would represent an improvement on the petrifying gaze which is currently a very short-lasting entangle effect that can be broken out of by physical strength (as a side note, I should have other limits piled on to that but the limit that the target has to be able to see covers most things). A transform physically transforms the target and requires them to spend a long time recovering, just as if they had been injured. The Partial Transform advantage allows for the effects to ramp up to the final level. For example, if the first use only gets a small number of points on the target, their skin might start turning grey but they wouldn't have real mechanical penalties yet.
One thing to note is the things that didn't improve. Basic combat values didn't improve even though archery specifically improved tremendously. Health and Endurance didn't improve. Basic characteristics didn't improve. Only the things that the character felt were lacking or which they were interested in pursuing. This fits in very well with the fact that the initial system started with superhero comics which is a fiction mode where characters appear to be more static in terms of power. They are often fighting the same enemies over and over again. They are always taking on the same rogue's gallery of minions and masterminds and when they gain new things it is mostly lateral or personality growth rather than pure vertical growth.
The growth seems to be slow, but even 1 character point can provide something that adds interesting things to a character. Also, psychologically, you tend not to notice the growth as much. A lot of the developments you get over time are taken to address specific problems and often don't come up regularly. Then suddenly the situation is back and you remember, oh, I can do this thing that deals with this. A lot of the time, the same basic thugs or orcs or the like you fought at the beginning remain just as dangerous, but now you have a broader range of options for how to deal with them. By comparison, a higher level D&D character tends to just flat out overpower the things they had difficulty with at 1st through 5th level.
One of the downsides of this is that you don't have those moments of suddenly displaying a massive boost of power. It can happen, but it takes a certain amount of planning as you have to set aside points and save them to spend all at once. This usually happens when you are saving up to buy one large trait.but there's nothing stopping you from saving and spending a whole bunch of points on a variety of characteristics or skills at once to simulate some sort of level-up or shonen "I'm more powerful than ever before" reveal scene.
It should also be noticed that while Hero System suggests for GMs to set caps on how powerful individual statistics can get, but there is no cap on the total point value of a character. Also, the GM can always decide that the scope of the campaign has increased and thus raise the caps of characteristics. Even in this case, unless the characters are faced with enemies that just flat blow them out of the water in the first encounter, they're rarely going to spend experience on improving the core central stats because you don't spend points for no reasons and if the character is still hitting and dealing damage reliably there is no reason to increase those capabilities. If the character still manages to get through fights with most of their Stun and Body, there's no need to improve defenses or Health. You spend points on interesting things and to address troubles.
Character Creation and the Gorgon Archer