pr.probability - Characteristic operator

The wikipedia page cited in the question provides most of the answer: to get your operator compute
begin{equation}
lim_{delta rightarrow 0} frac{ {mathbb E}[f(X_delta)] -f(x)}{delta}
end{equation}
The difference between your problem and the case covered in the wikipedia article is that $f$ in the above display is a function of $x$ only. However, your problem has an additional state variable (the binary variable that takes one of the values $1$ or $2$ depending on $alpha$). So, the correct limit to study is:
begin{equation}
lim_{delta rightarrow 0} frac{ {mathbb E}[f(X_delta,alpha_t)] -f(x,1)}{delta}.
end{equation}
Thus, you don't have one function, but two functions $f(x,1)$ and $f(x,2)$ and two PDEs that these functions satisfy.

It is implicitly assumed that $tau$ is independent of the dynamics of $X$ before $tau$. Furthermore, before $tau$ the dynamics of $X$ are governed by the first SDE given in the question. One can use these to write the above expectation in two pieces: one piece over the set ${delta < tau}$ the other over ${delta < tau}^c$. Once this is done, the usual use of Ito's formula gives:
$$
L_1 f(x,1)=-lambda f(x,2)~~~ (*)
$$
and
$$
L_2 f(x,2) = 0.
$$
where
$$
L_i = a(b_i - x) frac{partial}{partial x} + frac{1}{2} sigma^2frac{partial^2}{partial x^2}
$$

Further details:
begin{align*}
{mathbb E}[ f(X_delta,alpha_delta) ]&=
{mathbb E}[ f(X_delta,alpha_delta) 1_{{ tau > delta}} ] +
{mathbb E}[ f(X_delta,alpha_delta) 1_{{ tau le delta} }]\\
&approx (1-lambda delta){mathbb E}[ f(X^1_delta,alpha_delta)] +
lambda delta f(x,2),
end{align*}
where $X^1$ is a process that is independent of $tau$ with dynamics determined by $L_1$.

Here you use several things: 1) $P( tau < delta) approx delta lambda$ 2) if a jump occurs before $delta$, you can ignore what happens between $tau$ and $delta$ (the contribution of this part is in the order of $delta^2$ and when divided by $delta$ and
$delta$ is let go to $0$, it disappears).

To get (*) from the previous display: use Ito's formula on the first expectation, subtract $f(x,1)$, divide by $delta$ and let $delta rightarrow 0$. $f(x,2)$ is a function of what happens after $tau$; after $tau$ the stochastic process is a simple diffusion with generator $L_2$: this is why (**) holds.