How was this conditional probability decomposed? On this webage concerning the likelihood function of the Kalman filtering, the following lines of equations can be found:
\begin{align}
P[y_2\mid y_1]&=\int P[y_2,x_2\mid y_1]dx_2\\
&= \int P[y_2 \mid x_2]P[x_2 \mid y_1]dx_2
\end{align}
For some reason, I cannot replicate the second line - i.e.,transition from $P[y_2,x_2\mid y_1]$ to the product $P[y_2 \mid x_2]P[x_2 \mid y_1]$. I tried playing around with Bayes Theorem to no avail. How was this derived?

In general, you have:
$$
P[y_2, x_2\;|\; y_1] = P[y_2\;|\; x_2, y_1] P[x_2\;|\; y_1],
$$
which follows from the definition of conditional probability.
But for Kalman filters, the observation $y_2$ is independent of the observation $y_1$ given the state $x_2$, i.e.:
$$
P[y_2\;|\; x_2, y_1] = P[y_2\;|\; x_2].
$$
This results in your identity:
$$
P[y_2, x_2\;|\; y_1] = P[y_2 \mid x_2]P[x_2 \mid y_1].
$$
