First of all let me mention that I'm a physics student and therefore cannot tell you whether Lie theory is interesting for a mathematician. Also I don't know the specific book you are referring to. However, I can tell you why Lie groups and Lie algebras are one of the most essential pieces of maths that a theoretical physicists needs.
Lie groups and algebras appear throughout physics to mathematically implement the concept of symmetry. For example, physics should be invariant under translations in space and in time, spatial rotations and boosts. Spatial rotations are described by the rotation ("special orthogonal") group SO(3), spacetime rotations, that is spatial rotations and boosts, are described by the bigger Lorentz group SO(3,1) and if you add the spacetime translations you end up with the Poincaré group. All of these groups are Lie groups. The Poincaré symmetry leads to the very way how you write down physical laws, using so-called Lorentz indices (covariant notation).
In quantum field theory and elementary particle physics you encounter even more Lie groups describing so called internal symmetries. For example, the internal symmetry group of the standard model describing the strong and electroweak interaction is SU(3)xSU(2)xU(1) where SU(n) and U(n) are again Lie groups. Such symmetries help you to write down the theory and also work with it, i.e. do the spontaneous symmetry breaking using the Higgs mechanism which leads to the breaking of an initial internal symmetry group to a smaller one. The so called grand unified theories for example start with a big group like SU(5) and break it down to the standard model group SU(3)xSU(2)xU(1) using the Higgs mechanism.