Imagine you have a gadget --- like a polymer, kind of --- that admits as a configuration of itself any fixed-length "walk" to and fro along the real line, taking N steps of length Δx, each one either to the left or the right. Supposing that this walk starts at the origin, the total number of configurations that end up at point n looks like a Gaussian distribution: there's a hell of a lot more walks that end up close to where they started. So the polymer "wants" to be in a scrambled, balled up configuration, more so than it "wants" to be stretched out, because there are many more ways for it to be scrunched up than stretched out.
If you do a little thermodyamics back-of-the-napkin calculation, you find that apparently it does actually behave like a honest-to-goodness Hooke's-law-satisfying spring. Its "spring constant" turns out to be 2kBT/(Δx2N) where kB is Boltzmann's constant and T is the temperature in Kelvin. Or if you forget about the particulars of N and Δx and just think in terms of σ the standard deviation of the Gaussian distribution of possible displacements from one end of the polymer to the other, the spring constant is the beautifully simple kBT/σ2
So Boltzmann's constant is kind of a constant of proportionality between entropic forces and real forces, I guess?