For the exact use, history and significance of Laplace transform you may follow the Wikipedia. It will provide you a lot of information regarding your query.
In this connection, I would like to state that (as far your given information that you are a student of mathematics with no physics or engineering background) Laplace transform is a very powerful mathematical tool applied in various areas of science and engineering. It is frequently used in engineering and physics. In mathematics, it is particularly useful in solving linear ordinary differential equations. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. However, it also does provide qualitative information on the solution of the ordinary differential equations.
For more details you may find the reference:
"Laplace Transforms and Their Applications to Differential Equations" by N. W. McLachlan
Also you can follow the given two answer
What exactly is Laplace transform?
Applications of the Laplace Transform