Suppose $xy=N=\sqrt{N}\sqrt{N}$. If $x\ge\sqrt{N}$, then $y\le\sqrt{N}$ and vice-versa. Thus, if $xy=N$, then one of $x$ or $y$ must be less than or equal to $\sqrt{N}$. This means that if $N$ can be factored, one of the factors must be less than or equal to $\sqrt{N}$.
This is the contrapositive of what Gadi A said, but sometimes, if a statement doesn't make sense to you, its contrapositive might.
To answer the question asked: if you've crossed out the multiples of all the numbers less than or equal to $\sqrt{N}$, all multiples of numbers greater than $\sqrt{N}$ will already be crossed out. This is because any number which is less than or equal to $N$ and is a multiple of a number greater than $\sqrt{N}$, will have a factor that is less than or equal to $\sqrt{N}$ and therefore will already be crossed out. (Of course, when we are speaking of multiples here we mean multiples of $2\times$ or more, as in the Sieve.)