Learn Finnish: Kasvi kukoistaa, koska multa on kosteaa. - The plant flourishes because the soil is moist.

Because the subject it describes, multa (soil), is an uncountable noun (a mass noun). In Finnish, when you describe an uncountable noun using the verb olla (to be), the adjective must be in the partitive case. If you were describing a countable thing, like a towel (pyyhe), it would be nominative: pyyhe on kostea.

I've seen "multa" used to mean "from me" or "from another". Is that what it means here?

Good eye! While multa is the spoken Finnish version of minulta (from me) and the partitive form of muu (other), here it is a completely different word. This multa is a noun in its basic nominative case meaning "soil" or "earth".

The verb "kukoistaa" ends in a double vowel. Is it already in the third-person singular form here?

Yes, in this sentence it is the third-person singular ("it flourishes"). The dictionary form (infinitive) is also kukoistaa (to flourish). Because it's a Type 1 verb, the stem is kukoista-, and to form the "hän/se" form, you double the final vowel. This makes the conjugated form look exactly like the dictionary form!

Does the verb "kukoistaa" have an etymological connection to "kukka" (flower)?

Yes! Kukoistaa originally means "to bloom" or "to blossom", which comes directly from kukka. Just like the English word "flourish" (which comes from the Latin word for flower), it is widely used to mean that a living thing is thriving or doing exceptionally well.