When is the phrase, “Are you sitting down?” used, and what does it exactly mean?

Yes, that's what it means: "be prepared for some bad or, paradoxically, very good news I'm about to break." It's an introduction to the news someone is about to break over the phone and a reference to the possibility they may faint or have any sort of untoward symptom when they experience intense surprise, fear or shock. If one is sitting down, they are less likely to get hurt.

Examples of cases where you'd ask "are you sitting down?" before breaking the news:

• "Honey, I've just been held up and the son of a bitch ran away with all my money and credit cards!"
• "I'm pregnant and you are the father, obviously."
• "Remember that old coin you gave Jimmy? It was valuable and he sold it for $300."
• "Mom, I got full scholarship to Harvard Medical School."
• "Mike and I got married in Vegas yesterday."
• "The doctor said she has cancer."
• "We'll have to cancel our trip to Hawaii. Mr. Clyne, my boss, says..."

As we can see from the examples, it has more to do with "intense surprise" than with the news being good or bad.

When is the phrase, "Are you sitting down?" used,

When you are about to tell someone shocking news.

and what does it exactly mean?

It means, "I'm about to tell you bad news." It implies, "If you aren't now sitting, please sit down."
It can be taken literally, and probably should, to avoid risk of injury from fainting at the bad news, or jumping and hitting one's head.

For a literary-graphic illustration of exactly this, listen to Ed Gavagan's Moth story "Golf Clubbing" (7min 44sec) about (spoliers hidden)

injuries sustained to him (if you faint easily, sit down for it), his drill-sergeant father on learning the news, and his mother (who was, fortunately, seated) on learning the news about his father.

Having heard this phrase for years, this story was the best illustration I'd heard of it.
(Adult content warning: not for the queasy; occasional uncensored obscenity.)