Learn Russian: Студенты уснули от скуки и усталости. - The students fell asleep from boredom and fatigue.

This perfectly demonstrates the current lesson! We use от + Genitive for an involuntary physical or emotional cause. The students didn't choose to fall asleep as a conscious motive (which would take из). The sleep simply overtook them as a physical reaction to the boredom and fatigue.

Why is the verb уснули used instead of спали?

Уснули is the past tense of the perfective verb уснуть, which specifically means "to fall asleep" (a single, completed change of state). Спали is from the imperfective verb спать and means "slept" or "were sleeping." The sentence focuses on the event of them nodding off, not the duration of their sleep.

Both скуки and усталости end in -и. Do they follow the same declension rule?

They actually arrive at the -и ending via two different rules, though both are in the Genitive case because they follow the preposition от. Скука (boredom) is a feminine noun ending in -а, which changes to -и because of the 7-letter spelling rule (you cannot write 'ы' after 'к'). Усталость (fatigue) is a feminine noun ending in a soft sign (-ь), and all such nouns take an -и ending in the Genitive case.

Is усталости related to the phrase (I am tired)?