Learn Hardcore Latin: Puer clavem invenit et matri dicit se eam in via invenisse. - The boy finds the key and tells his mother that he found it in the street.

Because clavem is the accusative singular of clavis, clāvis (key). It’s the direct object of invenit (found), so Latin uses the accusative:

puer (subject, nominative)
clavem (object, accusative)

What case is matri, and why is it used?

matri is dative singular of māter, mātris (mother). The dative is used for the indirect object, i.e. the person to whom something is said:

matri dicit = he says to (his) mother

Why does Latin say matri instead of something like ad matrem?

Latin commonly uses the dative with verbs of speaking/telling to mean to someone, without a preposition. ad + accusative can mean to/toward, but it often emphasizes movement toward or direction, not simply the recipient of speech. Here, matri is the standard choice.

What is going on with se eam ... invenisse? Why isn’t it just ?