A quick Google should point you in the direction of what people use lambda calculus for:

http://en.wikipedia.org/wiki/Lambda_calculus

One fascinating area for the use of lambda calculus is as a conceptual basis for the implementation of functional programming languages - indeed you could argue that a language is functional iff it can be mapped to the lambda calculus.

Edit: Although it is rather old (like me!) I have fond memories of "The Implementation of Functional Programming Languages" by Simon Peyton Jones:

http://research.microsoft.com/en-us/um/people/simonpj/papers... [PDF]

Another very cool thing is that you can translate expressions in the lambda calculus to SK[I] combinators - which means you can actually implement real programs (including recursion - e.g. using the Y-combinator) as two extremely simple functions - not very efficiently, mind you... :-)

http://en.wikipedia.org/wiki/SKI_combinator_calculus

Lambda calculus is a simple model of substituting values into variables. For anyone studying the behaviour of such a system (eg. programming languages or Mathematical proofs) then LC is an nice, simple model to work with.

A physical-world analogy would be the billiard-ball model of atoms, which works well to describe the behaviour of gasses.

You're thinking about it from the wrong perspective, the lambda calculus is essentially the equivalent of assembly language for functional programming. The problems you can solve by building abstractions on top of it are infinite, it's utility comes from all properties we can deduce about it when building higher level abstractions, not from it not from being immediately useful to people programming at a higher level any more than x86 assembly is.