Learn Hardcore Ukrainian: Тобі варто виходити раніше, щоб встигати без поспіху. - You should leave earlier in order to make it without rushing.

Because варто is an impersonal predicate meaning “it’s advisable/it’s worth [doing].” Ukrainian uses the dative to mark the person to whom the recommendation applies (the “dative of experiencer”): Тобі варто… = “For you, it’s advisable to…”.
Other dative pronouns you’ll hear:

Мені варто… (for me)
Йому/їй варто… (for him/her)
Нам/вам варто… (for us/you-pl/polite)
Їм варто… (for them)

What’s the nuance of варто compared with треба / потрібно / слід / мусиш / повинен?

варто: soft, polite recommendation (“you’d do well to…”).
треба / потрібно: neutral necessity (“it’s necessary/you need to…”).
слід: “ought to,” somewhat formal/bookish.
мусиш / повинен: strong obligation (“you must/you’re obliged to”). Your sentence is advice, not an obligation, so варто fits well.

Does варто always need a person in the dative?

No. You can give general advice with no person: Варто виходити раніше… (“It’s advisable to leave earlier…”). Adding a dative (“Тобі варто…”) personalizes who the advice is for.

Why is the verb (imperfective) used here instead of (perfective)?