Is there any particular reason to why the Queen needs to be "Saved"?

It was written at least as far back as 1745, when it was much more about the monarch than the people. The phrase “God Save The King” appears in the King James Bible, which was completed in 1611. As early as 1545 “God Save the King” was a watchword of the Royal Navy, with the response being “Long to reign over us”. It’s a fairly new concept that things are about the people and not the monarch or leader of whatever kind, so it makes sense. The first definitive publishing of it was in 1744, before America was even a country. Many nations use the tune as their anthem and simply re-write the lyrics to suit their nation.

You have to understand that at one time it was illegal to think ill-wishes at the royalty. It has to do with magical thinking. It was believed that people could make things happen just by thinking about it. If you harboured bad thoughts about someone, that person might well take sick and die as a result. This wasn’t just superstition, this was regarded as fact. The opposite was also therefore held to be plausible, that enough people thinking well of the royalty could keep them healthy and hale. Speaking ill of the Crown was a crime which could be punishable by death. Hence the phrase, “The King is dead, long live the King!” You don’t want to risk negative thoughts of a dead King from afflicting the new King.

(This same sort of magical thinking persists into our modern era, incidentally. There is no evidence that comics books, pornography, trashy detective novels, video games, or any of the other stuff reactionaries like to rant about causes any actual harm; the magical thinking is that if a person imagines violence for long enough, it will somehow magically come to pass.)

Save in this instance just means keep safe.

God save the Queen
The fascist regime,
They made you a moron
A potential H-bomb.

God save the Queen
She ain’t no human being.
There is no future
In England’s dreamland

Don’t be told what you want
Don’t be told what you need.
There’s no future
There’s no future
There’s no future for you

God save the Queen
We mean it man
We love our queen
God saves

God save the Queen
‘cos tourists are money
And our figurehead
Is not what she seems

Oh God save history
God save your mad parade
Oh Lord God have mercy
All crimes are paid.

When there’s no future
How can there be sin
We’re the flowers
In the dustbin
We’re the poison
In your human machine
We’re the future
You’re future

God save the Queen
We mean it man
We love our queen
God saves

God save the Queen
We mean it man
There is no future
In England’s dreamland

No future
No future for you
No future for me

Satan.

To save means, in this instance, to keep from harm. (Safety derives from the same root.)

A dictionary is a marvelous invention. See sense 2.

We use the word in a similar sense (though not exactly the same) all the time: “Aren’t you going to finish your sandwich?” “I’m saving it for later.” Preserving it. There’s no implication of danger.

Because it makes a cool name for an epic sword.

It is used to refer to the institution of the royalty more than the person themselves, since before England became a Constitutional Monarchy the loss of the king or queen would throw the country into diplomatic turmoil as potential successors fought over the throne, and it would open up the country to invaders. The monarchy needs to be saved to protect the people and the country, and to keep everything in order.

The history isn’t as simple as one guy with a pen. Wiki has a nice long history of the phrase and song in this article. @Jeruba summed up the use of the word ‘saved’ nicely.

And even though it isn’t strictly helpful, this is good for a chuckle:
…It’s God Save The Queen. The Queen lives in a big house with barbed wire and people with guns. That’s one saved fucking Queen. She’s overly saved, she has no idea of the struggle of human existence. We have to work, raise a family. We don’t have nannies. It’s what you gotta do in your life. God Save The Queen, no.

God Attack The Queen, that’s what it should be.
“God attack the Queen, send big dogs after her, that bite her bum. Let them chase after her and rip her knickers off…?”
That’d be fantastic. She’d have to fight the dog, with a handbag with a brick inside.

“Crazy dog!”
“Rrrargh! Kill the Queen.”
“No, crazy dog!”

Maybe she’d kill the crazy dog, and everyone in Britain would go,
“Fair play to the Queen.”
The Queen would have self-respect for the first time in her life. Yes. lt would work. It would be fantabulous.
~Eddie Izzard

if she died the people would have no leader?

@wwwlover no Charles would be the leader (or more acurately Head of State). Depending on what Regnal name he takes he could be Charles III, although he himself has sugested that he may take Geroge VII.