Learn Finnish: Minusta tulee uuden alan asiantuntija. - I will become an expert in a new field.

This is the classic transformation construction you are learning in this lesson. Instead of saying "I become X", Finnish phrases it as "Out of me comes X". The person or thing transforming is put in the elative case (ending in -sta or -stä), which gives us minusta.

If minusta refers to me, why is the verb tulee in the "he/she/it" form instead of matching "me"?

In this sentence structure, the grammatical subject is actually the result of the change (the expert, asiantuntija), not you. Think of the literal translation as: "Out of me comes an expert." Because "an expert" is a third-person singular noun, the verb takes the third-person singular form tulee.

Why is the result (asiantuntija) in the nominative (basic) form? Shouldn't a change of state use the translative case?

You can use the translative case for a change of state, but it requires a different sentence structure: Minä tulen asiantuntijaksi. In the specific "Elative + tulee" structure used in this sentence, the result must always be in the nominative case.

Why are uuden and alan in the genitive case?