Learn Italian: Il fiore esotico, il cui seme è raro, sboccia in primavera. - The exotic flower, whose seed is rare, blooms in spring.

It matches the seed! When using cui to mean "whose", the definite article right before it must agree in gender and number with the noun being possessed (the seed, il seme), not the owner (the flower, il fiore). Because seme is masculine singular, we use il cui.

How do we know if seme is masculine or feminine since it ends in '-e'?

Nouns ending in '-e' can be either masculine or feminine, so you usually just have to memorize their gender. Seme is masculine. Luckily, in this sentence, the relative pronoun il cui gives you a massive hint: that il tells you seme is masculine singular!

Why is it just in primavera instead of nella primavera (in the spring)?

In Italian, seasons generally take the preposition in without a definite article. So you will normally say in primavera, in estate, in autunno, and in inverno. You only add the article if you are specifying a particular spring, like nella primavera del 2020 (in the spring of 2020).

What is the infinitive of sboccia?

The infinitive is sbocciare, a regular -are verb meaning "to bloom" or "to blossom". While it's most commonly used for plants, it can also be used figuratively for a person's talent or a new relationship.