You program in it! Take a look at church encodings. You can see how pretty much all arithmetic can be performed which should probably convince you that it is extremely powerful. I like to look at operations on lists however. You can define most any data structure in terms of a function that does the most important operation on it.
For instance an encoding of a list is the fold function that folds over it. Note this is not Church's encoding but one I got from Percie's types and programming languages. Church's pair encodings do not give us recursion we have to add it back in ourselves with some kind of recursion combinator.
so a list takes two arguments, a function to do the folding, and a initial value to plug into the fold at some point.
cons x xs = lam f. lam a. f x (xs f a)
nil = lam f. lam a. a
now we can define a summation given an add function (see church encodings from above)
sum xs = xs add 0
we can do more and define a map function
consApply f x xs = cons (f x) xs
map f xs = xs (consApply f) nil
if you are still not convinced that there is computation going on here and want to make sure that you can perform any computation then check out the fixed point combinator. It hurts my head a bit to think about sometimes however so I'm not sure I would call it intuitive but if you manually evaluate it with some arguments you can see what is going on.