Ég þarf að hvíla mig eftir fundinn.

Because þurfa normally takes að + infinitive.

So:

• ég þarf = I need
• að hvíla = to rest

Together, ég þarf að hvíla means I need to rest.

You cannot normally leave out að here.

Hvíla is the infinitive, the basic dictionary form of the verb.

After að, Icelandic usually uses the infinitive:

• að hvíla = to rest

So this is similar to English to rest.

Because Icelandic often uses a reflexive idea here: literally, to rest myself.

So:

• hvíla mig = rest myself
• natural English translation: rest

English usually does not say I need to rest myself in this situation, but Icelandic commonly uses the object pronoun.

Because the subject is ég (I), so the matching object form is mig (me).

• ég = I
• mig = me

Sig is a reflexive form used for third person subjects, like he/she/they in certain contexts, not for I.

So:

• Ég þarf að hvíla mig = correct
• Ég þarf að hvíla sig = incorrect

Because mig is the object form, while ég is the subject form.

Compare:

• Ég þarf... = I need...
• ...hvíla mig = ...rest me/myself

English also changes pronouns this way:

• I see me
• ég sé mig

Because fundur is the basic nominative form, but after eftir in this sentence, the noun appears in a different case.

Here:

• fundur = meeting (basic form)
• fundinn = the meeting in the form required here

So eftir fundinn means after the meeting.

In Icelandic, the definite article is often attached to the end of the noun instead of being a separate word.

So:

• fundur = a meeting / meeting
• fundinn = the meeting (in this case form)

That -inn ending is the attached the.

Eftir means after here.

So:

• eftir fundinn = after the meeting

It is a preposition, and like Icelandic prepositions generally, it affects the form of the noun that follows it.

Yes, that is also possible.

• að hvíla mig = to rest myself
• að hvílast = to rest

Both can be natural, and learners will encounter both patterns. The sentence you were given uses the straightforward verb + object pronoun pattern.

Yes, Icelandic word order is somewhat flexible.

The given sentence is a very neutral way to say it:

• Ég þarf að hvíla mig eftir fundinn.

But you could also say:

• Eftir fundinn þarf ég að hvíla mig.

That puts more focus on after the meeting. Both are natural, but the original version is a very standard default order.

Þ is pronounced like the th in thing, not like the th in this.

So:

• þarf begins with the unvoiced th sound: tharf

This letter is one of the most distinctive features of Icelandic spelling.