Anyone who knows anything is envied by someone 
Anyone who knows anything is envied by someone.

Let


*

*$K (x,y) = x \text{ knows } y$

*$E (x,y) = x \text{ is envied by } y$
I feel like anyone who knows "anything" is supposed to refer to anyone who knows everything, but apparently I'm wrong? Please help!
 A: 
  
*
  
*"Anyone who knows anything is envied by someone." 
  

This indeed does not claim only that anyone who knows everything is envied.
Rather it states that: If there is a knows relation between any $x$ and any $y$, then there is an envied-by relation between that $x$ and some $z$.


*

*"For any $x$ and any $y$, there is some $z$ that: if $x$ knows $y$, then $x$ is envied by $z$."


$$\forall x~\forall y~\exists z~\big(\mathrm K(x,y)\to \mathrm E(x,z)\big)$$

PS: amWhy suggests an equivalent statement:


*

*"For any $x$, if there is some $y$ known by $x$, then there is some $z$ that $x$ is envied-by".


$$\forall x~\big((\exists y~\mathrm K(x,y))\to(\exists z~\mathrm E(x,z))\big)$$
A: This is more of an English question than a mathematical one. English uses "any" to quantify at the outermost level. In contrast, "every" is used to quantify at that point. Hence:

If you know anything then ...  ≡  For any thing x, if you know x then ...
If you know everything then ...  ≡  If ( for every thing x, you know x ) then ...
I do not know anything  ≡  For any thing x, I do not know x

There can still be ambiguity...
"I do not know everything" could mean either of:

I do not ( know everything )  ≡  It is not true that ( for every thing x, I know x )
I ( do not know ) everything  ≡  For every thing x, I do not know x

But generally you would not go wrong to interpret based on the distinction I stated, using context to resolve ambiguities.
This also explains why "any" sometimes means "some" but at other times means "every". For instance your sentence:

Anyone who knows anything is envied by someone.  ≡ 
Everyone who knows something is envied by someone.  ≡ 
For any person p and any thing x such that p knows x, there is some person q such that p is envied by q.   (as in Graham's answer)

Often we can use "some" or "every" to make the logical meaning of "any" clear. But sometimes, ambiguity with "everything" (due to being able to refer to the entire collection) can be removed by judicious use of "any":

I do not know anything  ≡  For any thing x, I do not know x.   [no ambiguity!]

A: "Anything" requires care in quantifying, and your confusion about why anything should mean everything, that's good intuition.  Because anything doesn't always mean everything. 
What we want to say is something of the form: given all people $x$, if there exists something $y$ such that $x$ knows $y$, then $x$ is envied by someone $z$.
$$\forall x\Big(\exists y\big(K(x,y)\big) \to \exists z\big(E(x, z)\big)\Big)$$
Domain of discourse of $x$, $z$: all people.  Domain of $y$: all things that can be known.
A: Note that the contrapositive :


*

*$(P\to Q)\equiv (\lnot Q\to\lnot P)$


Here is quite easy to formulate: 


*

*if you are envied by no one it's because you know nothing

Transcribed: $\forall x\left(E(x,\varnothing)\to K(x,\varnothing)\right)$
Since the negation of $\varnothing$ is $\exists y\quad $ (belonging to an implicit set $Y$ not named, e.g. other people, knowledge).
You get the initial statement after contraposition: $\forall x\left(\exists y\,K(x,y)\to\exists z\,E(x,z)\right)$ 
as stated by amWhy.
