How was Le Chatelier's principle used to determine the extinction coefficient?

There is only one Le Châtelier's Principle, whereas there are many extinction coefficients, as many existing coloured substances have their own extinction coefficient and can be involved in several equilibrium reactions.

There are general experimental and mathematical approaches to the general problem, but in case of a specific equilibrium reaction and coloured substance that you are implicitly referring to, it would be easier to tailor the explanation/help to that case.

So, you should talk about the task to your teacher and then, after diving into the problem, pose a more specific question about it to "Socratic".

You may have been trying to determine the formation constant of a complex ion like the thiocyanatoferrate(III) ion.

Fe³⁺ + SCN⁻ ⇌ FeSCN²⁺

The reaction does not go to completion, because this is an equilibrium process.

You probably added a large excess of the SCN⁻ ion to Fe³⁺ solutions of different concentrations.

According to Le Châtelier's Principle, this shifted the position of equilibrium far to the right.

The number of moles of the complex ion formed were then essentially equal to the initial number of moles of Fe³⁺ ion.

Thus, your different dilutions of the Fe³⁺ ion had known concentrations of the complex ion.

You could then make a Beer's Law plot to determine the extinction coefficient of the FeSCN²⁺ ion.