I was reading Cohn's book on Lie Groups.In introduction part he has given the motivation behind Lie Groups.It is like this
If solution of the differential equation $\frac{dx_{i}}{dt}=u(t)$ is $x_{i}=f_{i}(x,t) $ then we can think of this solution as a linear transformation $x'=S_{t}(x)$ such that each point $x$ can be associated with a new point $x'$ reached after time $t$ .Then family of transformations $(S_{t})$ on the whole space forms a group,with the operation $S_{t}.S_{t'}(x)=S_{t+t'}$.
I could not understand the whole concept of constructing transformations from differential equations and the defined operation