Learn Russian: Он хорошо относится к детям.

In English, you "treat" someone directly. In Russian, относиться literally means "to relate" or "to have an attitude." To express how you treat someone, Russian phrasing is like saying how you "relate towards" them. This requires the preposition к (towards) followed by the Dative case.

Since our current lesson focuses on verbs that take the Dative without a preposition, why is there a к here?

Verbs of communication (like promising or answering) take a bare Dative because they represent giving a message to an indirect object. However, verbs describing an emotional attitude or behavior directed towards a target, like относиться, require the preposition к (towards) alongside the Dative case to show where the attitude is aimed.

What is the base form of детям and why does it look so different?

The word for "child" is highly irregular. In the singular it is ребёнок, but the plural base changes completely to дети (children). детям is the Dative plural form of дети, taking the standard -ям ending for soft-stem plurals.

Why does хорошо come before the verb instead of at the end of the sentence?

In Russian, adverbs of manner like хорошо typically come immediately before the verb they modify. While putting it at the end wouldn't be grammatically wrong, хорошо относится is the standard, natural word order unless you specifically want to emphasize the word "well" by moving it to the very end.