Learn Russian: Моему дедушке под восемьдесят, но он хорошо спит. - My grandfather is nearing eighty, but he sleeps well.

You're right that под literally means "under." However, when combined with a number representing age or time, it takes the Accusative case and translates to "approaching," "getting on for," or "nearly." It implies the person is getting very close to that milestone.

Why is Моему дедушке in the Dative case?

In Russian, expressing someone's age always requires the person to be in the Dative case. The underlying literal structure is "To my grandfather is [approaching 80]." Even though the English translation uses "My grandfather" as the subject, Russian keeps its standard Dative age construction when using this new approximation rule.

Why isn't the word for "years" (лет) included after восемьдесят?

When using под to express an approximate age, the word лет is almost always dropped. The context already makes it clear you are talking about age, and omitting it sounds much more natural and idiomatic in conversational Russian.

The lesson says под takes the Accusative case for age, but восемьдесят looks like the Nominative case. Why is that?

It actually is in the Accusative case! For the numerals 50, 60, 70, and 80, the Nominative and Accusative forms are identical. So восемьдесят follows the grammar rule perfectly; it just doesn't change its spelling.