Learn Hardcore Icelandic: Ég borða morgunmat á hverjum degi.

Because morgunmat here is the accusative singular (the direct object of borða). The dictionary form (nominative) is morgunmatur, but strong masculine nouns in -ur drop -ur in the accusative: matur → mat, so morgunmatur → morgunmat.

Why is there no article before morgunmat?

This sentence uses morgunmat in an indefinite sense (“(some) breakfast”), so no article is needed. Icelandic expresses indefiniteness by using the bare noun. To say “the breakfast” you add the definite suffix: morgunmaturinn.

What does á hverjum degi literally mean and why is it in the dative?

Literally it means “on each day.” The preposition á (“on”) takes the dative case when expressing time. Therefore hverjum is the dative of hver (“each”) and degi is the dative of dagur (“day”).

Can I say Ég borða morgunmat hvern dag or Ég borða morgunmat daglega?

Yes. hvern dag (“every day”) uses the accusative without a preposition, and daglega is an adverb meaning “daily.” Both are correct:

Ég borða morgunmat hvern dag.
Ég borða morgunmat daglega.

What happens if I put á hverjum degi at the beginning of the sentence?

Icelandic follows the V2 rule (the finite verb must be second). Fronting á hverjum degi gives:
Á hverjum degi borða ég morgunmat.

Can I drop the subject pronoun Ég?

Because borða is inflected for 1st person singular, you can technically omit Ég:
Borða morgunmat á hverjum degi.
However, in simple statements it’s more natural to include Ég for clarity.

What’s the difference between borða and eta?

Both mean “to eat.” Borða is the standard transitive verb, and eta is also used (eg ég et). You can choose either:

Ég borða morgunmat á hverjum degi.
Ég et morgunmat á hverjum degi.