Learn Icelandic: Ég afsaka þetta. - I apologize for this.

In English, we say 'apologize for' something, but the Icelandic verb afsaka takes the thing you are apologizing for directly as an object. It translates more literally to 'I excuse this'.

I usually hear afsakið for 'excuse me'. Is this the same word?

Yes, it is the exact same verb! Afsakið is the command form used politely to say 'excuse me'. In this sentence, afsaka is just the normal present tense form for ég (I).

Why is the word for 'this' þetta instead of þessi?

Þetta is the neuter form. In Icelandic, when you use 'this' to refer to a general situation, an action, or an abstract idea rather than a specific noun, you always use the neuter form.

What grammatical case is þetta in here?

It is in the accusative case because it is the direct object of the verb afsaka. For neuter words like þetta, the accusative form is identical to the nominative (subject) form, so it does not change its spelling here.