How does practicing restraint and performing actions of merit purify the mind?

Adhearing to the precepts will only have a positive effect on your mental states if you are currently breaking them. If the girl is always agitated because she is a chronic liar, the precepts will help her. If she is always agitated because she doesn't like her desk job, the precepts will not help her.

What the precepts attempt to prevent are common actions that lead almost always to distracting mental states.

The precepts are a guideline of things you can abstain from to make practicing the cultivation of concentration and wisdom easier. Your meditation sessions will not be as fruitful if you are constantly worried about getting caught in your lie.

loosely, as i understand it,

There are two things going on;

1. Behavioral conditioning; formation of inclinations and tendencies. The way this occurs can be explained in terms of formation, dismantling and the strengthening of neural formations.
2. Tweaking the range of possible experiences; consequent direction of one's reality to the supposedly desirable states which happen to have certain pre-requisite causes and conditions. The decisions we make direct us to certain experiences, thus if we aim at specific experiences we ought to bring about the requisite conditions to inevitably bring about the desired outcome.

When one is set on attaining the yet-unattained one usually relies on external guidance as to how to bring about the yet-unattained. In case of the precepts their purpose and result is non-regret, non-regret as ie regret of not following directions, here is an example;

'Person A' is convinced that there exists a 'Place' and that he needs to go to the 'Place' asap as it was a matter of life and death.

However he does not know where it is and how to get there.

'Person A' asks 'Person B' and is told the directions. 'Person A' directs himself to the destination by following the path as it was proclaimed.

Then 'Person A' gets an idea to deviate from the Path and subjectively speaking if his goal is to get to the 'Place' asap, if he was to deviate from the Path, it would absolutely compromise his objective and bring about regret and remorse for wrong-doing. It is wrong here because it is against the objective, if achieving the objective is desired then it is discerned as good, failure is discerned as bad, therefore what leads to failure is bad and what is bad is a cause for remorse and regret.

“Sir, what is the purpose and benefit of skillful ethics?” “Ānanda, having no regrets is the purpose and benefit of skillful ethics.”

“But what’s the purpose and benefit of having no regrets?”

“Joy is the purpose and benefit of having no regrets.”

“But what’s the purpose and benefit of joy?”

“Rapture …”

“But what’s the purpose and benefit of rapture?”

“Tranquility …”

“But what’s the purpose and benefit of tranquility?”

“Bliss …”

“But what’s the purpose and benefit of bliss?”

“Immersion …”

“But what’s the purpose and benefit of immersion?”

“Truly knowing and seeing …”

“But what’s the purpose and benefit of truly knowing and seeing?”

“Disillusionment and dispassion …”

“But what’s the purpose and benefit of disillusionment and dispassion?”

“Knowledge and vision of freedom is the purpose and benefit of disillusionment and dispassion.

“Mendicants, an ethical person, who has fulfilled ethical conduct, need not make a wish: ‘May I have no regrets!’ It’s only natural that an ethical person has no regrets. When you have no regrets you need not make a wish: ‘May I feel joy!’ It’s only natural that joy springs up when you have no regrets. When you feel joy you need not make a wish: ‘May I experience rapture!’ It’s only natural that rapture arises when you’re joyful. When your mind is full of rapture you need not make a wish: ‘May my body become tranquil!’ It’s only natural that your body becomes tranquil when your mind is full of rapture. When your body is tranquil you need not make a wish: ‘May I feel bliss!’ It’s only natural to feel bliss when your body is tranquil. When you feel bliss you need not make a wish: ‘May my mind be immersed in samādhi!’ It’s only natural for the mind to be immersed in samādhi when you feel bliss. When your mind is immersed in samādhi you need not make a wish: ‘May I truly know and see!’ It’s only natural to truly know and see when your mind is immersed in samādhi. When you truly know and see you need not make a wish: ‘May I become disillusioned and dispassionate!’ It’s only natural to become disillusioned and dispassionate when you truly know and see. When you’re disillusioned and dispassionate you need not make a wish: ‘May I realize the knowledge and vision of freedom!’ It’s only natural to realize the knowledge and vision of freedom when you’re disillusioned and dispassionate.

If she was only keeping precepts she would not experience the consequences of not following the precepts.

Even tho our reality is subjective and relative as in one's head is big in relation to the head of an ant but is small in relative to an elephant's head, there are also absolutes, ie if one's head is blown off to pieces conditions one will absolutely die.

Similarly one can know that eventho there are many different states called happiness and the discernment of happiness is subjective there is also absolute happiness and everyone can agree that not experiencing unhappiness is happiness and the escape from suffering is pleasant.

Therefore absolutely everyone is interested in the absolute removal of what can be discerned as gross in light of what is supreme.

Therefore when there are to that extent absolutes of good and bad, decision making associated with the ultimate good even at the level of mere morals and conventions will have higher expectation than the consequences of the unfavorable decisions.

The consequences will be as they will be regardless of the subjective perspective and one's understanding of them which may or may not be in accord with objective reality.

All of the three or ten kinds of merits have the function to straighten right view. Once a gross healing and cleaning has reached, right view could be attained, the good Dhamma heard, seen by one self and as a result the path would develop, once the Noble eightfold path is reached, it's fruits in gaining liberation will appear naturaly, on it's given cause.

And as mentioned in another answer, it's function is to gain a clean stand, good selfestimate, by cleaning and become rightly free of remorse, able to access borderlands to the Noble Domain. If not having done merits on what should one build on a good and required stand/state?

As the Buddha told, merits are that what beings establish in auspicious states, states where they could be able to access the path to freedom, go on beyound.

(Note: this is given for doing merits, not for trade, exchange or stackes to bind and stay bound)

was always agitated, resentful and frustrated

frustrated

I think "frustrated" might mean that you want to do something but can't. But keeping the precepts might be relatively easy, might be within reach, something anyone can do. It's not like "getting a job" or "getting married" etc. where you depend on someone else to give you something -- keeping precepts is something you can do by and for yourself. So you can do that, and a "success" is the opposite of a "frustration".

resentful

I think that "resentful" sounds like the classic, "He abused me, he struck me, he overpowered me, he robbed me." The precepts tend to concern social behaviour. I imagine that if you break the precepts -- kill people, steal from them, seduce their spouses or children, lie to them, and so on -- then they might retaliate in some way. If they punish you, get angry, stop trusting you, then you might resent that. Or the opposite, if you keep the precepts and people are friendly towards you, then you have less reason to resent them.

Keeping the precepts involves a certain amount of thinking about other people, of "Putting oneself in the place of another" -- I think that kind of unselfishness is an antidote to resentment.

agitated

It's less obvious that 'merely' keeping the precepts is an antidote to agitation -- I think it's possible to keep the precepts and to still experience agitation.

Keeping precepts might reduce the range of what's conceivable -- e.g. lusting after other people's things and bodies becomes off-limits -- maybe that reduces attachment and craving and gets you to think about (wholesome) desire instead.

Also the precepts are fairly stable -- maybe they're suitable as an object of contemplation, i.e. if you "keep them in mind" then that replaces agitation.

Also I'm not sure about "positive", but "less negative" seems obvious.

I don't know if you've met, known, or been an alcoholic for example, but the pursuit of alcohol seems to me a classic example of samsara, e.g. you have some alcohol, you like that, the effect wears off (is impermanent), so you want more. But, the habit is harmful in several ways, and is an activity which doesn't nourish you in a way that might be necessary for you to be more successful and more "positive".

Perhaps the same is true for the other pursuits too which are forbidden by the precepts, i.e. lies, killing, lust, theft.

One more thing is that I think that some of the precepts have further consequences.

The Ambalatthika-rahulovada Sutta (MN 61) for example,

In the same way, Rahula, when anyone feels no shame in telling a deliberate lie, there is no evil, I tell you, he will not do. Thus, Rahula, you should train yourself, 'I will not tell a deliberate lie even in jest.

That engages "shame" as a motive -- i.e. you avoid doing all sorts of harmful things, don't even consider them, to avoid shame.

And the next part of the sutta is abut self-reflection, "Is this action beneficial, or harmful? Will the consequences be pain, or happiness?" That's a kind of question with far-reaching consequences, and maybe it's important to try for 'honest' answers.

And not lying (to others) is an important component if you ever want to ask someone to support your withdrawing from an addiction.

There's a term called "habit energy", or "underlying tendency", the hidden force that influences our motive to do/speak/think in wholesome or unwholesome ways. As simple analogy is a chain smoker would tend to carry on his unhealthy habit due to it being persistently re-inforced for so long. But if s/he has the strong will to replace the bad habit with good habits of eating healthy, exercising frequently, etc., then overtime the s/he'll be able to build up a new wholesome "habit energy" strong enough to overpower the older unhealthy one. Practice senses restraint thru the Five Precepts are just like that.

It is mostly the opposite of

How would practicing the five precepts, make her calm and peaceful?

the result of sati sampajana is the respect of the precepts, not the contrary. This is what puthujjanas struggle a lot to understand. Any puthujjana can follow the 5 precepts for 5 minutes, but it is sati and sampajana a which makes this hold in the long term.

Well, it seems obvious to me now, but it wasn't always, that our immoral impulsive behavior, both of possessive type and of repulsive type, comes from emotional hangups, so giving in to those hangups feeds them, or creates circumstances that feed them indirectly. So, if this immoral behavior comes from the mind, then by managing behavior we learn to control what?

Anyone who tried the precepts knows they are impossible to keep without paying attention to one's mind and emotions. Precepts force us to control our external behavior, which requires controlling our mind, that's it. No magic.