A matrix is used to convert a vector from one coordinate system to another (say, from coordinate system 'A' to coordinate system 'B').
The inverse of a matrix is a matrix which converts the other direction (say, from 'B' back into 'A')
In games, we commonly have a matrix which will transform positions from world-space into homogenous eye-space, since rendering all happens in homogenous eye-space. But homogenous coordinates can be difficult to work with, so some types of shaders want to convert those homogenous camera-relative coordinates back into world coordinates, so inverting the matrix allows us to easily convert those coordinates back into world-space inside the shader.
In models we often have a hierarchy, where each piece of the model has a matrix which describes how to convert from its local coordinate system (used by its own vertices), to the coordinate system used by its parent, so that you can multiply a vertex's position by that matrix to find where the vertex is relative to the parent. (And if you then multiply by the parent's matrix, and the parent's parent's matrix, and so on, you'll eventually find the vertex's absolute world position) If you invert the child-to-parent matrix, though, it allows you to go the other way; take one of the parent's vertices, multiply its position by the matrix, and the result will tell you where that position would be, expressed inside the child's coordinate system (and thus, relative to its vertices).
In a game, we normally specify matrices going just one direction; from deeply nested renderable objects, through less-deeply-nested renderable objects, through a view matrix, and finally into homogenous eye-space for rendering. Which is normally all we need, since all we normally want to do is to find world-space or render-space. But for those unusual cases where we really do want to know where a world-space point is relative to a nested object (or which direction a light is shining relative to a movable object, or etc), then inverting matrices allows us to transform vectors between coordinate systems in the opposite direction.