How did "bail" shift to signify "money deposited as a guarantee when released"?

Perhaps most helpful to you would be to read the section on the word's etymology, where you will see that the original meanings are French - but the notion of bail as some form of surety appears to be entirely English.

Etymology: In senses 1, 2, < Old French bail ‘power, custody, jurisdiction,’ and ‘delivery,’ noun of action < baillier , in its senses of ‘take charge of, guard, control,’ and ‘hand over, deliver’ (see bail v.1). The remaining senses are peculiarly English, and their development not quite certain. But from the Latin phrase tradere in ballium alicui , found 1259, and the Anglo-Norman lesser en bail (1331), it seems that bail was originally related to sense 1, and meant the ‘custody or charge’ (i.e. of the surety); compare also the equivalent mainprise , manucaptio . But it must soon have been associated in meaning with sense 2, and the Anglo-Norman bailler to deliver (chattels), since already in 16th cent. it was explained as ‘the delivery or handing over of the accused to his surety,’ whence transferred to the security in consideration of which he was so delivered, and finally to the surety himself who became or gave security for his reappearance in court when called for judgement. There is nothing approaching these senses in French, while, on the other hand, the French sense ‘lease’ is not in English. French bail , the noun of action, must be carefully distinguished from bail ‘governor, ruler, tutor, guardian’ (= Provençal baile , Spanish bayle , Portuguese bailio , Italian bailo , balio , < Latin bājulus ), never adopted in English (though sometimes erroneously stated to be the direct source of sense 6). Bail=bājulus was the primary word; thence the verb bailler=bājulāre; thence again the noun of action bail formed in French, and adopted in English. (The word is discussed by Coke 4th. Inst. 178.)

Sense 1 of bail per the OED, and now said to be Obsolete has examples from 1400 to 1599.

Charge, custody, jurisdiction, power. Obsolete.

Sense 2 is also considered Obsolete.

Sense 3 concerns the friendly custody of a prisoner in lieu of imprisonment

All the other senses, i.e. 4 to 6, to my mind bear some resemblance to the way the word is used today - involving a surety in lieu of imprisonment.

This includes sense 5a, which has an entry from 1331 - albeit written in Norman French.

5a. Security given for the release of a prisoner from imprisonment, pending his trial.

[1331 Act 5 Edw. III viii Et ne les soeffrent nulle part aler walkerants ne par bail ne santz bail.]

1495 Act 11 Hen. VII c. 7 They [shall]..be put to sufficient baill.

1592 J. Stow Annales 580/1 Giuing liberty to..his prisoners to go vnder baile.

A “bailor” under the common law of contract is “one who takes possession of and responsibility for a piece of property, and is liable to the owner of the property (the bailee) if it is lost or damaged while it is in their possession. “Bail” applies that concept to criminal legal proceedings.

The person who posts bail for a defendant is pledging an amount of money in order in exchange for possession of a prisoner. The party posting the bail is the “bailor” and the criminal justice system is the “bailee.” The bailor has pledged that the defendant shall be returned to the court, and if the bailor fails to ensure the return of the defendant to court, then they surrender the security that they pledged to the court in satisfaction of the liability that they assumed when they took possession of the defendant.