There is actually a more general version of this theorem for any two finite fields of the same cardinality. But we'll stick to this baby version.
We can denote any such field as $F = \{0,1,a,b\}$.
A small detour:
The mapping $\sigma:F \to F$ given by:
$0 \mapsto 0$
$1 \mapsto 1$
$a \mapsto b$
$b \mapsto a$
is a field automorphism.
Since this is clearly bijective, we only need to show it is a ring homomorphism.
To do this, we need to know what the addition and multiplication tables for $F$ actually are. Sums involving 0 are clear, so we need not investigate those.
A small detour to our detour:
Claim: 1 + 1 = 0.
Since $(F,+)$ is an abelian group, we must have either:
1 + 1 = 0, or 1 + 1 + 1 + 1 = 0. Suppose it is the latter. Then $1 + 1 = a$ (or $b$, but since we haven't specified anything about $a$ or $b$, we'll just say it's $a$). It is then clear that $1 + 1 + 1 = a + 1 = b$.
Now by the distributive law (since we're in a field):
$a\cdot a = (1 + 1)(1 + 1) = (1)(1 + 1) + (1)(1 + 1) = (1 + 1) + (1 + 1) = 1 + 1 + 1 + 1 = 0$.
Thus $a$ (which is distinct from 0) is a zero divisor, which cannot happen in a field. This proves our claim.
Claim 2: $x + x = 0$, for all $x \in F$.
Proof: $x + x = x(1) + x(1) = x(1 + 1) = x(0) = 0$ (the proof that $x(0) = 0$ is easy. Do it).
So the additive group $(F,+)$ is isomorphic to $\Bbb Z_2 \times \Bbb Z_2$, under addition modulo 2. This completely determines the addition: for example: $a+1$ cannot be 0 ($a$ is its own additive inverse), it cannot be 1 (since that implies $a = 0$), and it cannot be $a$ (since that implies $0 = 1$ which cannot happen in a field), so $a + 1 = b$.
Since this is "one less letter", we will henceforth write $a+1$ instead of $b$.
Which leads us to multiplication. The products involving 0 or 1 are trivial to define, and I will not bother. That leaves (because of commutativity of the product), just 3 products to investigate:
$a\cdot a$
$a \cdot (a+1)$
$(a+1)\cdot(a+1)$
By the distributive law, once we know $a \cdot a$, we'll know the rest. Now since $(F-\{0\},\cdot)$ forms an abelian group (because we have a field), and this group has 3 elements, it is clear that it is a cyclic group of order 3 (since all groups of order 3 are, since 3 is a prime), so any non-identity element is a generator. This means that:
$a\cdot a \neq 1$, because $a$ must be of (multiplicative) order 3, not order 2. If:
$a \cdot a = a$, then multiplying by $a^{-1}$ yields $a = 1$, which we know isn't true. Thus $a \cdot a = a + 1$.
Now it should be clear that the map $\sigma:F \to F$ above yields a GROUP automorphism of $(F,+)$ (we're just "switching two of the generators of the Klein 4-group"). So we have a FIELD automorphism if and only if $\sigma$ is a group automorphism of the multiplicative group of $F$. But on $F-\{0\}$, the map $\sigma$ is just the inversion map:
$x \mapsto x^{-1}$ (can you see this?)
which is a homomorphism because the multiplicative group is abelian.
Ok, so if we have a second field $F' = \{0',1',a',a'+1\}$, since any ring-homomorphism $\phi$ must take:
$0 \to 0'$
$1 \to 1'$ we either have:
$\phi(a) = a'$
$\phi(a+1) = a'+1$
which is clearly a ring (and thus field) isomorphism, or:
$\phi(a) = a'+1$
$\phi(a+1) = a'$
in which case $\phi \circ \sigma$ is clearly a field isomorphism and thus so is:
$\phi = (\phi \circ \sigma) \circ \sigma^{-1}$
(the composition of two isomorphisms is an isomorphism).