Learn Russian: Из-за тревоги мне совсем не спится. - Because of anxiety, I can't sleep at all.

This is the core of the impersonal involuntary state you are learning! While я не сплю just states the factual action that you are not sleeping, мне не спится means that you want or are trying to sleep but cannot. Using the Dative pronoun мне and the reflexive verb спится shows that the situation is out of your control, as if sleep just won't come to you.

Why does the sentence use из-за for 'because of'? Could I use потому что?

Из-за is a preposition used directly before a noun to indicate a cause, very often a negative one or a barrier. You cannot use потому что here because it is a conjunction that must be followed by a complete phrase with a subject and a verb, such as потому что я тревожусь (because I am anxious).

Why is the word тревога changed to тревоги here?

The preposition из-за always requires the noun that follows it to be in the Genitive case. Since тревога (anxiety) is a feminine noun ending in -а, its ending changes to -и in the Genitive case.

How exactly does совсем work with the negative verb in this sentence?