Think of it as a game: the first player gets to pick any $x$ at all, and the second player then has to find a $y$ that makes the statement $x=2y$ true. If the second player can always do this, the quantified statement $\forall x\exists y(x=2y)$ is true; if she can’t always do it, the quantified statement is false.
If we’re talking about integers, it’s false: if the first player picks $x=1$, the second player won’t be able to find an integer $y$ such that $2y=1$. If we’re talking about the real numbers, on the other hand, it’s true: no matter what $x$ the first player picks, the second player just chooses $x/2$ for $y$.
Now look at the statement with the quantifiers in the opposite order: $\exists y\forall x(x=2y)$. This time it’s the first player who is picking something, namely a number $y$, and he wins if no matter what $x$ the second player chooses, that $x=2y$. Equivalently, the second player wins if she can find an $x$ that makes the statement $x=2y$ false. And of course she always can, no matter what $y$ the first player chose, whether we’re talking about integers or about real numbers. In either domain the quantified statement $\exists y\forall x(x=2y)$ is false.
On the other hand, $\exists y\forall x(x+y=x)$ is true in both domains: the first player just has to pick $y=0$.