Lie theory is, among other things, a very elegant way to parameterize an energy function involving rotations in 3D space. For example the translation and rotation of multiple camera frames that observe the same set of 3D points. This is the basis of how Visual Odometry, SLAM or 3D reconstruction problems are solved.

There are other cool things like representing a probability distribution over a 3D rotation with an multivariate gaussian vector interpreted as a so3 or se3 vector. Using this you can for implement a very elegant extended kalman filter estimating a 3D rotation. Which usually can be a tricky thing to do.

In robotics we pretty much only use the SO3, SE3 and SIM3 groups so my knowledge is limited to those groups.