I do not think it is particularly profitable to name a single application or even several applications; rest assured that there are many such applications.
Let me make a more general point instead. One caricature of how mathematics is done these days is that it is done by examining the kinds of structure that appear in mathematical situations. Structure can take many forms and there are many names we give to particular kinds of structure:
- group structure,
- ring structure,
- order structure,
- topological structure,
- metric structure (this is one of the kinds of structure that real analysis in particular deals with)
etc., and the history of mathematics is full of surprises about what kinds of structure can appear in a particular mathematical situation. (For example, an astonishing variety of different kinds of structure - algebraic, topological, analytic, etc. - appear in modern number theory.) Moreover, seemingly different kinds of structure interact in useful, surprising, and beautiful ways. To do mathematics in this way it is therefore vital to have a good grasp of at least the basic kinds of structure that have repeatedly proven to be indispensable in mathematics.
Real analysis, from one point of view, studies metric structure. This is fundamental: it will probably be your first rigorous introduction to the absolutely crucial ideas of limits and approximation, without which, for example, the entire discipline of science would be meaningless (if our physical theories do not approximate nature then they are useless). These crucial ideas recur all over mathematics and real analysis is only your first opportunity to begin wrestling with them.