"Fresh Apples here! Only the best apples for 2 dollars!" -
"I would like one, please." -
"Thatll be 2.50, sir." -
"What? I thought you just said 2?" -
"Demand has just gone up."
If you go to a store, the store owner sees you take your apples up to the counter, and so he can theoretically change his price before you get there (although in practice, if he ever did that he would soon be out of business).
On a financial exchange, the market maker doesn't even find out that you wanted to buy until the trade has already happened. It is literally impossible for the market maker to change his price, because he doesn't find out about your order until it's already occurred.
What is possible is that the market maker is also quoting on another, totally separate exchange, and he decides to change his prices there, in reaction to seeing a big order on the first exchange.
It's like an apple seller who owns two carts in different parts of town. When you come to his first cart and buy all his apples for $2, he guesses that maybe you are going to go over to his second cart and buy all the apples there as well, so he calls his business partner who's running that cart, and tells him to raise his prices to $2.50 - which makes perfect sense as a business strategy, because demand has gone up.
Note that he only raised his prices because you bought all the apples at his first cart. If you just bought one apple out of the hundreds he has (because you're a small investor, not a giant investment bank) then he wouldn't bother to raise his prices.
Then again, I have no idea about these things and should probably shut up.
"someone wants to buy a number of shares at a quoted price"
"a giant investment bank or hedge fund with some privileged
information about a stock wants to buy so many shares that
they actually need to go to multiple exchanges to satisfy
their demand"
A much better one is to think of a string of gas stations running down the highway. They all have an advertised price. A tanker truck arrives at the first gas station and buys all the gas at the advertised price. It then goes down the road and buys all the next stations gas at the same advertised price. Then the manager of the 2nd gas station calls the 3rd and says hey, we've just gotten wiped out of gas there is a lot of demand. The 3rd manager raises his price accordingly. The tanker truck can then decide if it wants to buy more gas at the new price or just take what they currently have.
"Fresh Apples here! Only the best apples for 2 dollars!" - "I would like every single apple you have, please. Also I'm buying all the apples from the guy across the street too." - "Thatll be 2.05 each, sir." - "What? I thought you just said 2?" - "Demand has just gone up."