The DA decides
The decision to prosecute a crime is vested in the district attorney (a.k.a. the prosecuting attorney, a public official in charge with bringing criminal cases in court).
The DA can prosecute someone even if the victim doesn't press charges.
It isn't the victim's call.
It isn't law enforcement's call (although the DA will never get to make that decision if neither the victim nor law enforcement tell the DA about it).
Critically, in U.S. law, unlike most countries with "civil law" legal system in Continental Europe and Latin America and much of Asia and Africa, prosecutors don't have a legal duty to prosecute all crimes of which they are aware. Civil law systems sometimes give vetoes over prosecutions of particular kinds of crimes to victims, but the general rule is that a prosecutor must press all charges that can be proved and that the prosecutor has resources sufficient to prosecute. In the U.S., the decision to bring charges or not is in the absolute discretion of the prosecutor.
Also, unless the suspect and the DA reach an express agreement to permanently drop the charges, the DA or a successor to the DA can change their mind at any time until the statute of limitations on the crime expires.
In the example in the question, a DA might initially decide not to bring charges, but then change her mind and bring the charge from this incident and a lot of other ones, when the DA learns that the suspect is operating a full fledged criminal enterprise and not just exercising bad judgment on an isolated basis.
The DA usually honors a victim's wishes
Usually, a DA and law enforcement will honor a victim's wishes, both because it is the harm to the victim that the DA is primarily vindicating, and because a case can be hard to prosecute without the victim's cooperation.
Lots of people are pressing to have scarce law enforcement and DA resources applied to their problems. When someone voluntarily withdraws their request to have the DA and law enforcement use those resources, and no one personally involved is unhappy about that, this is normally seen as a win for everyone, and as a way of empowering victims.
DAs and victims alike are also well aware that a criminal case can have very severe impacts on the life of the criminal defendant and the criminal defendant's family. Sometimes it can literally ruin a person's life. Other times it presents a major bump in the road to someone who was overall getting on the right track but had a lapse of judgment. When deciding whether to bring charges, DAs routinely weigh whether the harm done by breaking the law justifies the consequences of bringing a criminal case against someone from the point of view of society as a whole in the long run.
In this example, the victim got moral vindication, an apology and admission, and presumably, his stuff back. The victim and law enforcement clearly saw this as justice enough if this is really just an isolated incident in a case involving a fairly minor misdemeanor offense.
Exceptions to the general rule
In the unusual case where a DA brings a criminal case notwithstanding the victim wanting to drop the charges a variety of things can motivate that.
Sometimes, for example, in an intrafamily domestic violence or child abuse or elder abuse scenario, the DA may conclude that the victim is asking to drop charges under duress, or figurative "temporary insanity" due to being blinded by love in a way that the victim will later regret.
In a variation on this fact pattern, fraud victims often have lawsuits for compensatory damages against fraudsters that will be harder to collect if the fraudster is in prison and may ask to drop charges for that reason. But the DA may want to bring the criminal fraud case anyway, because it is likely that the future income that will used to pay the current victims from their lawsuit will just come from some new fraud perpetrated against someone else.
Sometimes, a DA presses charges because even though the victim doesn't care, the DA or law enforcement believes that the criminal is a high risk for being a repeat offender. This is especially true in cases where the collateral consequences of a conviction (e.g. disqualification from possessing a firearm) may help to prevent a future crime.
Also, keep in mind that the DA is either an elected official, or a political appointee, or reports to someone who is. Usually, victims who get what they ask for from the DA build up political support. But, sometimes, a criminal case will be popular with the general public to prosecute even if the victim doesn't want that to happen. So, that is another reason that a DA might prosecute a case over the victim's objections.
Conclusion
Still, the basic thing to keep in mind is that prosecutors bringing charges over the objections of the victims are the rare exception, because prosecutors do care about what victims want, and to some extent see victims as proxies for being their clients, even though their true client is the government or the abstract concept of "the People."
Also, victims who don't want to press charges are themselves pretty rare.
