Ég er mjög þreyttur, svo ég fer að sofa strax.

I

Breakdown of Ég er mjög þreyttur, svo ég fer að sofa strax.

• vera: to be
• mjög: very
• strax: immediately

Questions & Answers about Ég er mjög þreyttur, svo ég fer að sofa strax.

Why is þreyttur ending in -ur? What if the speaker is a woman?

Þreyttur is an adjective agreeing with the subject ég in gender, number, and case. Here the implied speaker is masculine singular nominative, so you get þreyttur.
If the speaker is a woman, you’d normally say Ég er mjög þreytt (feminine singular nominative).
(Neuter would also be þreytt, e.g. Barnið er þreytt.)

Why is it Ég er and not Ég er að vera or something else?

To describe a state like being tired, Icelandic uses the verb vera (er = present tense “am/is/are”) + an adjective: Ég er þreyttur/þreytt.
You don’t use a progressive construction here; að vera would be the infinitive “to be” and wouldn’t fit this structure.

What does mjög do grammatically? Does it change form?

Mjög is an adverb meaning “very” that intensifies an adjective or another adverb. It does not decline or change form:

• mjög þreyttur = “very tired”
  You keep mjög the same regardless of gender/case/number.

Why is there a comma before svo?

Because you have two independent clauses: 1) Ég er mjög þreyttur
2) (…) ég fer að sofa strax
The comma helps separate them when svo is used like “so/therefore” to connect two full clauses.