Physically, what does the usage of two variables mean for convolution
My intuition of convolution is that it is just a way to depict multiplication of two signals where each signal is made up of various frequencies and phases.

Not really, no.  I can't think of a case where you'd actually receive two signals and convolve them (correlate them -- yes, but not convolve).  This may be why you're not understanding the rest.
Convolution is used to find the output of a linear system* when you know the system's impulse response.

And intuitively, convolutions are seen as sliding windows.  ... $\tau$ is very clearly some dummy variable but beyond that I feel a little confused.

$\tau$ is the dummy variable that implements the sliding window in the convolution.  For each time $t$, you multiply the signal by the time-reversed impulse response, and you integrate that product.  $\tau$ is just the variable over which you do the integration.
* I purposely left the &quot;time-invariant&quot; part of that out -- convolution applies to time-varying systems, too, it's just more complicated.