Question: A morphism $F$ between algebraic varieties $V\subseteq \Bbb{A}^n$ and $W\subseteq \Bbb{A}^m$ is the restriction of a polynomial map on the ambient affine spaces $\Bbb{A}^m$ and $\Bbb{A}^n$. 

What does this mean? Say we have the mapping $F:\Bbb{A}^2\to\Bbb{A}^3$ defined by $(x,y)\to (x+y,x^2+y^2,x^3+y^3)$. Also, we choose $\Bbb{V}(x-y)$ to be the variety in $\Bbb{A}^2$ and $\Bbb{V}(x+2y+3z)$ in $\Bbb{A}^3$. Does $F$ necessarily map $\Bbb{V}(x-y)$ to $\Bbb{V}(x+2y+3z)$? What is happening?

Answer: 
Let me just set up notation again. I've got $F\colon \mathbb{A}^n \to \mathbb{A}^m$ and affine varieties $V \subset \mathbb{A}^n$ and $W \subset \mathbb{A}^m$. When is it the case that $p \in V$ implies $F(p) \in W$? The condition to be in $W$ is that $g(F(p)) = 0$ for each $g$ in some set of polynomials cutting out $W$, so I want the polynomial function $g \circ F$ to vanish on $V$; ie, it should lie in $I(V)$.

In your example it might be better to give different names to the coordinates on the source and target, but forging on we check that $x + 2y + 3z$ pulls back to
\[
(x + y) + 2(x^2 + y^2) + 3(x^3 + y^3).
\] 
This is not a multiple of the generator $x - y$ of $I(V)$. Of course, just checking that $F(1, 1) = (2, 2, 2)$ doesn't satisfy $x + 2y + 3z$ is a fine argument as well.
Your example: F(1,1)=(2,2,2)\not \in V(x+2y+3z). So F does not send a arbitrary closed set in any arbitrary closed set. There are easier examples: Take a very small closed in $\mathbb{A}^3$ like $V(x,y,z)$ and notice that most polynomials map most closed sets in $\mathbb{A}^2$ outside this closed set. 

I find that statement vague. 
By definition, a morphism of varieties is a map that pullbacks regular functions, locally a quotient of polynomials, to regular functions. A globally defined regular function on an affine variety $W$ is just an element of the coordinate ring $A(W)$ of $W$. 
Let $W\subset \mathbb{A}^n$ be the ambient affine space. Then $A(W)=A(\mathbb{A}^n)/I(W)=\frac{k[x\_1,\ldots,x\_n]}{I(W)}$, where $I(W)$ is the ideal of vanishing functions on $W$.

Also, a map $F:V\rightarrow W$ between varieties is a morphism of varieties if and only if $\varphi\_i = x\_i \circ F$ is a regular function for all $i$ on $\mathbb{V}$. Here $x\_i$ are the coordinate functions sending a tuple $(z\_i)$ to its ith component $z\_i$.
For your function $x\_1 \circ F= x+y$, $x\_2 \circ F= x^2+y^2$. 

Note that $F(P)=(\varphi\_1(P),\varphi\_2(P),\ldots)$.

Combining these, a function between affine varieties is called a \emph{morphism} if and only if its components are globally defined regular functions
Globally defined regular functions on an affine are polynomials on the ambient space viewed modulo some ideal.

So one could say morphisms of varieties are restrictions of polynomial maps on the ambient spaces..

See [Ben Moonen - Algebraic Geometry Notes](http://www.math.ru.nl/~bmoonen/AlgGeom/alggeom.pdf "Ben Moonen - Algebraic Geometry") Chapter 2.