(heavy spoilers ahead - read at own risk)
As we know the events around the first prophecy of Trelawney go like this:
- Dumbledore interviews her
- She makes the prophecy
- They are interrupted by Snape who overhears part of the prophecy
- Snape rushes to Voldemort and reports what he did overhear
- Meanwhile Dumbledore decides to hire Trelawney (although this might be just for her safety) and let children waste their time with a pointless subject
- Voldemort tries to kill Harry but fails
Here is the quote from Harry Potter and The Half Blood Prince chapter 25:
'Please let me finish.' Dumbledore waited until Harry had nodded curtly, then went on. 'Professor Snape made a terrible mistake. He was still in Lord Voldemort's employ on the night he heard the first half of Professor Trelawney's prophecy. Naturally, he hastened to tell his master what he had heard, for it concerned his master most deeply. But he did not know - he had no possible way of knowing - which boy Voldemort would hunt from then onwards, or that the parents he would destroy in his murderous quest were people that Professor Snape knew, that they were your mother and father -'
Why does everyone - Snape, Dumbledore and Voldemort take this prophecy so seriously?
We have multiple examples where Voldemort does not take it easy if someone wastes his time with pointless news - the punishment for that is heavy. So Snape must have been 100% sure that what he overheads is important.
How was Snape sure that the whole thing wasn't a fraud from Trelawney to get the job?
Voldemort also takes everything very seriously and starts hunting down the Potters with the idea to do something that is evil even by his standards - to kill a toddler. Again based on what? On something that one of his servants heard and nothing more?
Dumbledore seems to think that the prophecy is actually not relevant by itself and becomes important BECAUSE Voldemort acted on it:
"Yes — just love," said Dumbledore. "But Harry, never forget that what the prophecy says is only significant because Voldemort made it so. I told you this at the end of last year. Voldemort singled you out as the person who would be most dangerous to him — and in doing so, he made you the person who would be most dangerous to him!"
"But it comes to the same —"
"No, it doesn't!" said Dumbledore, sounding impatient now. Pointing at Harry with his black, withered hand, he said,
"You are setting too much store by the prophecy!"
"But," spluttered Harry, "but you said the prophecy means —“
"If Voldemort had never heard of the prophecy, would it have been fulfilled? Would it have meant anything? Of course not! Do you think every prophecy in the Hall of Prophecy has been fulfilled?"
This is of course Dumbledore's guess but we know that his guesses are usually true :)
So besides being a major plot driver - is there any in-universe explanation about the behaviour of the people who heard the prophecy.