All substitutions in mathematics are typically used to take one representation to another that is more useful or easier to use in the context of the problem.
In terms of DE's, the goal is to find the unique set of functions that satisfy the DE. It may be a family functions or it may be a unique one.
At the end of the day, you are trying to take a DE and find a function representation satisfying that. What the DE corresponds to and why its useful is usually answered by physicists and other scientists, or applied mathematicians. The theoretical/pure side is interested in getting a solution, knowing a proper solution exists, and knowing some of its other properties (like convergence, differentiation, etc).