Learn Icelandic: Hans grimmi dreki borðar kjötið.

The possessive word hans (his) triggers the weak form of the adjective. For masculine, singular, nominative nouns like dreki (dragon), the weak adjective ending is -i.

I thought possessives usually go after the noun, like dreki hans. Why is hans at the front here?

You are right that possessives usually follow the noun in Icelandic! However, when you add an adjective into the mix, the possessive often moves in front of the adjective and noun. So instead of dreki hans, we say hans grimmi dreki.

Both grimmi and dreki end in -i. Do weak adjectives always match the noun's ending perfectly like this?

Not always! It happens to match here because dreki is a weak masculine noun (which ends in -i) and the weak masculine adjective ending is also -i. But if the noun were feminine or neuter, the weak adjective ending would be -a, which might not match the noun's ending.

What case is kjötið (the meat) in here?

It is in the accusative case because it is the direct object being eaten. Because kjöt is a neuter noun, its accusative form looks exactly the same as its nominative form. The -ið at the end is the attached definite article.