Different force constant in ground vs excited state in spectroscopy
At the risk of providing a somewhat circular answer, in the harmonic approximation of an interatomic potential, the force constant is a measure of the force of attraction between the atoms (in a classical description they would obey Hooke's Law $|\vec{F}|=-k(x-x_\circ$)). The answer to the question is therefore that the attractive interatomic force in the electronically excited state of $\ce{I2}$ is weaker than in the ground state.

The following figure from Ref. 1 illustrates the potentials in the ground and excited states: 



The potential in the excited (upper) state is broader. In the Morse functional approximation, the potential has the form $$V(r)=D_e\left(1-e^{-\beta(r-r_\circ)}\right)^2$$

where the width of the potential is proportional to $\beta$ and $$\beta = \left(\frac{k}{2D_e}\right)^{1/2}$$

A smaller force constant results in a smaller value of $\beta$ and a broader potential well. This fits nicely with the results noted in the OP and the above figure (the equations are also outlined in the accompanying text). The energy accompanying a vibrational transition is proportional to the square root of the force constant and therefore inversely proportional to the width of the potential well. This is analogous to the behavior observed for a QM particle in a box, wherein the spacing between energy levels narrows as the box size increases. Working backward from vibrational transition energies leads to the conclusion that smaller transition energies are associated with greater oscillatory amplitudes (broader potentials) and smaller harmonic force constant estimates.

References


Bayram, S.B. and Freamat, M.V., A Spectral Analysis of Laser Induced Fluorescence of Iodine. arXiv eprint 1507.02600, 2015.