Learn Russian: Мой день рождения в мае.

Мой agrees with the main noun in the phrase, день (day), which is masculine. The word рождения is just describing the day. So you are literally saying "My day of birth."

Why is the word for birth (рождение) in the genitive case (рождения) here?

Russian doesn't have a single word for "birthday." Instead, you use the noun phrase день рождения, which literally translates to "day of birth." The genitive case is used to show this "of" relationship.

The lesson talks about using the genitive case for months. Why is в мае in the prepositional case?

We use the genitive case of a month when giving a specific date, like "the 5th of May". But when you just want to say something happens in a month, without a specific day, you use the preposition в plus the prepositional case (в мае).

How does the spelling of май change to become мае?

The nominative (dictionary) form of the month is май, which is a masculine noun ending in -й. To form the prepositional case, we drop the -й and add -е, resulting in мае.