Converting a non-stationary random process into a WSS process by adding a random phase Here is an example where this method has been implemented.

We were trying to calculate the spectrum of a transmitted signal(Random signal/weighted pulse)


The auto correlation function of the pulse which is not WSS because it is dependent on t:





We associated a random variable(which is the delay) with an uniform PDF with the signal: 





PDF of the delay/the variable is:


 

I was unable to grasp the concept behind it, probably my basics are a bit rusty.


By simply just associating a random variable (with an uniform PDF), how can we just make any random process a wide sense stationary process?
Whats the concept behind it?
It is not true that you can convert any non-stationary random process into a WSS random process by just adding a random phase. What is true is that you can make a (wide-sense) cyclostationary process WSS by adding a random phase. A common example is the PAM process mentioned in your question:

$$x(t)=\sum_ka_kp(t-kT)\tag{1}$$

The discrete values $a_k$ in $(1)$ are assumed to be random. The process $x(t)$ can be shown to be cyclostationary. The modified process

$$\tilde{x}(t)=\sum_ka_kp(t-kT+\theta)\tag{2}$$

where $\theta$ is a random variable uniformly distributed on $[0,T]$ is WSS. This random phase reflects our uncertainty about the phase of the signal, i.e., about the origin of the time axis. In this sense, it is physically motivated, and not just a nice "trick" enabling us to compute the power spectrum of $(2)$.