What is the physical significance of the universal gas constant R?
It may be helpful to look at a related value $k_\mathrm{B}$, the Boltzmann constant, which is widely used in thermodynamics.
These two are related by $R = k_\mathrm{B}N_\mathrm{A}$, allowing the ideal gas law to also be written:
$$PV = Nk_\mathrm{B}T$$ where $N$ is the number of particles, as opposed to the number of moles.
The units are $\pu{J\cdot K^{-1}}$.
It's a proportionality between energy and temperature. In the ideal gas law, it relates temperature on the right to the product of pressure and volume, which is an energy.
This general idea is frequently used in thermodynamics, as you will see factors of the form $\exp{(-E/k_\mathrm{B}T)}$, where the $k_\mathrm{B}$ allows the exponent here to be unitless.
As examples:

Planck's law, where the energy is in the form of quantum energy level spacing:
$$B_{\nu}(\nu, T) = \frac{2h\nu^{3}}{c^{2}}\frac{1}{\exp\left(\frac{h\nu}{k_{\mathrm{B}}T}\right) - 1}$$
Maxwell-Boltzmann distribution, where the energy refers to the kinetic energy of gas molecules:
$$f(v) = \left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)^{\frac{3}{2}}4\pi v^{2}\exp\left(-\frac{mv^{2}}{2k_{\mathrm{B}}T}\right)$$