You'd explain it the same way you'd explain it to a fifty year old person.
What about this...
"As you know, there are different colours ... red, green and so on. The air likes to bounce around these colours. Amazingly enough, the air most likes to bounce around blue. The other colours don't get bounced around as much ... they just go away. So when you look up at the sky, we see all the blue being bounced around. The air is bouncing the blue around!"
(Older children may then ask, "Awesome! But why does the air bounce around blue and not some other colour?" The best answer then is something like, "It's because of the different size and speed of the different colours. It turns out that air is best at bouncing around blue. The others just pass through - they go away. Look up at the sky, and you see all the blue bouncing around!")
Further, for three year olds, there's nothing at all wrong with introducing the colours as "little balls"1 within this explanation, if desired.
Looking to the other superb answers, also talk about how big the air is. (It's possible that three year olds know what the "space station" or "satellites" are these days, with endless Netflix and iPad. Using the techniques in the other answers, you could show how much air there is, "all the way to the space stations!" "where it is black" because there is "no air to bounce the colours around" up there.)
Secondly from the other answers, you can make the excellent point about sunset. 781
"When you're looking more at the sun, at sunset, the red balls make it through the air to you - the air bounces away the blue balls." (Of course, this is getting less and less precise - let us say more and more 19th century - by and by.)
So that's how you explain the sky being blue to anyone, including three year olds. You explain it by explaining how it is. The air bounces blue around. But first!...
But first!!!!
Critically, as a pedagogical matter I would preface the whole explanation like this: this is more important than the actual explanation.
"That is a REALLY HARD question. It's important that you know, nobody had a clue about this until very recently. Like, when your grandmother {whatever} was alive. There was this great man called <PERSON> who figured it out* {I'm not totally sure if three year olds know from TV the cliché of "<PERSON>" as our great scientist - if so, bingo, that's good enough}. Before that it was really a mystery. It's still a mystery."***
I'm not quite sure of the best way to express it to a three year old, but it is critical to impress on the three year old, in terms they can understand, that
* humans presently know almost nothing
* it's only since the Enlightenment - very recently - that we have tried to figure things out. (Give a quick synopsis of the Enlightenment, starting with the renaissance in Italy)
* don't forget we only had electricity, phones, visited the moon, etc very recently ("you're grandmother got them")
* many many questions you can ask are very very difficult - your Mom doesn't know the answer, even your Aunty who works at the Big School with the 30 km tunnel underground doesn't know the answer
* when you're a little older, you, too, can start trying to figure out these questions, along with Mr <PERSON>, your Aunty etc
This "little speech" will dwarf in life significance any explanation of why the sky is blue. (BTW asking "why the sky is blue" is astounding for a three year old, I'm sure the kid is almost four!)
Trials
Unfortunately I don't have a three year old on hand at the moment - that is awfully young - but testing shows this is perfectly understandable to 7, 8 year olds.
The bottom line three year old explanation is: "air likes to bounce around the different colors. But mostly it likes to bounce around blue. So when you look up at the air, you see all this blue."
(The other colours "pass through" or "go away" ... "pass through" is too complicated a concept for three year olds, so wing it there.)
1 given that currently, our base understanding of the fundamental nature of energy and matter is: Utterly Nothing Whatsoever: there's nothing wrong at all with talking about "little balls!" to a THREE year old! 781