Learn Icelandic: Hann meiðir hnéð í fótbolta. - He hurts his knee playing football.

In Icelandic, when you talk about things happening to body parts, you usually just use the definite article instead of a possessive word like hans. Adding the -ð to hné makes it "the knee", and the context makes it clear that it is his own knee.

Doesn't the verb meiða (to hurt) usually take the reflexive pronoun sig?

Yes, if you want to say "he hurts himself" it would be hann meiðir sig. However, because this sentence names the specific body part that is hurt (hnéð), that body part replaces sig as the direct object of the verb.

Why does the English translation say "playing football" when the Icelandic just says í fótbolta?

Icelandic commonly uses the preposition í (in) plus the name of the sport to express playing a game. So í fótbolta literally means "in football", which is the most natural way to describe participating in a sport in Icelandic.

What grammatical case is fótbolta in this sentence?

It is in the dative case. The preposition í takes the dative case when it describes an ongoing state or activity, like participating in a sport. The dictionary form of the word is fótbolti, which changes its ending to an -a in the dative.