The goal of this passage in the textbook is to establish the following fact:
Fact 1: Suppose $K_1$ and $K_2$ are covariance functions (i.e., symmetric, positive definite, functions). Then $K_1 K_2$ is also a covariance function.
Their argument is this: suppose $f$ and $g$ are independent Gaussian processes with kernels $K_1$ and $K_2$ and mean function $m(x) = 0$. Then $f(x) g(x)$ is a stochastic process with covariance function $K' = K_1 K_2$. This is relatively easy to check just by direct calculation, and all we are really using is the fact that there exist mean-0 stochastic processes with the associated $K_1$ and $K_2$. The assumption that $f(x)$ and $g(x)$ are GPs is actually superfluous, it's just convenient to use them because you already know such an $f$ and $g$ exist.
The second point they make is the following:
Fact 2: There exists a Gaussian process with covariance function $K_1 K_2$.
Now, just because there exists a Gaussian process with this kernel, does not imply that $f(x)g(x)$ is a Gaussian process. We only considered this product to establish that $K_1 K_2$ is a valid covariance function. But, in general, if you take a product of Gaussian random variables, there is no reason to expect that the product will still remain Gaussian; for example, if $X \sim N(0,1)$ and $Y \sim N(0,1)$ then $XY$ is not Gaussian anymore. But, as long as $K_1 K_2$ is the covariance function for some stochastic process, it will immediately follow that there is a Gaussian process (different from the stochastic process we started with) which has this covariance.