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I have been told that whenever I'm drawing a circuit in the frequency (s) domain, the voltage source is to be replaced by V/s and the current source by I/s.

But what exactly is the reason?

When I'm plotting this circuit (replacing s with rad/s), I get that the voltage across R is decreasing as the frequency increase. intuitively, I would say that the voltage or current source should only have a frequency component at s=0, and 0 everywhere else, instead we have that as s approaches 0, the value of the voltage or current source approaches \$ \infty \$ .

schematic

simulate this circuit – Schematic created using CircuitLab

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It's because the Laplace transform is: $$A \to \dfrac{A}{s}$$

You assume that the source is applied at t=0: $$\int_{0}^{\infty} A e^{-st} \, dt = \frac{A}{s}$$

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