Well, sometimes it's hard (or annoying) to find solutions to ODEs by hand and, hence, it is easier to show that there exists a (unique) solution to an initial value problem depending on the information you're given.
However, in a lot of cases, existence/uniqueness is established in an attempt to formulate a general setting in which you are guaranteed to have a solution to an ODE, which is unique.
For example, the classical analysis problem: Show that there exists a solution to y'=f(x,y), y(0)=y_0, \|f(x,y)-f(x,z)\| \le k\|y-z\| in some interval [0,a], where x\in[0,a], y:[0,a]\rightarrow\mathbb{R} and f \in C([0,a]\times\mathbb{R}).
This could be formulated as a fixed point problem-- i.e. fixed points of:
Ty=y_0 + \int_{0}^{x}f(t,y(t))dt, x\in[0,a], would satisfy the ODE above.
Hence, one may use the Banach contraction mapping principle after showing that Ty is a contraction mapping (in particular instances), and hence conclude that there exists a unique solution y to the problem Ty=y and hence, existence and uniqueness is established for this class of ODEs. (You would end up with some sort of restriction on a though)
I have no idea why you would be exposed to these sorts of ideas without proofs, since that would sort of defy the purpose.
