Learn Russian: Он осторожно прикоснулся к её руке.

The prefix при- often indicates an action that is slight, incomplete, or gentle. Here, it combines with the root for touching (-кос-) to mean a very light or brief contact. This softening effect matches perfectly with the adverb осторожно (carefully).

Why do we use the preposition к and the dative case (руке) instead of just an accusative direct object?

In Russian, verbs of touching, leaning, or joining that start with при- usually require the preposition к followed by the dative case. Instead of directly acting upon the object, the grammar frames it as directing your movement or contact towards (к) the object.

I noticed an -о- in the root of прикоснулся. Isn't the root for touch normally -кас-?

Yes, but this root has a predictable vowel alternation: -кас- versus -кос-. The imperfective form of this verb is прикасаться with an -а-, but because this sentence uses the perfective past tense, it switches to -о- in прикоснулся.

Could I use потрогал instead of прикоснулся in this context?

You could, but it changes the nuance. Потрогать means to actively touch, feel, or handle something, like checking if a radiator is hot. Прикоснуться specifically means to make light, often momentary contact, which fits the gentle tone of this sentence much better.