Learn Icelandic: Mig langar að vera heima í kvöld.

The verb langa (to want) is special. Instead of using the normal subject word Ég (I), it forces the person who wants something into the accusative case, which is Mig (me). Think of it as literally saying "It longs me".

If I am the one wanting, why doesn't the verb change to match a first-person "I" form?

Because Mig is in the accusative case, the verb doesn't recognize it as a normal subject. Whenever a verb has one of these "quirky" subjects, it locks into its default 3rd-person singular (he/she/it) form. For langa, that form is always langar.

Why do we need the word "að" before "vera"?

When you follow langar with an action, you use að to connect them. It works just like the English word "to" in "want to be", marking vera as the infinitive (dictionary) form of the verb.

I learned the word for home is "heim". Why is it "heima" here?

Icelandic has different words for locations depending on whether you are moving or staying still. You use heim when you are traveling there (motion), but you use heima when you are already staying in that place, like in this sentence.

Does "í kvöld" literally translate to tonight?