How exactly does lambda calculus capture the intuitive notion of computability? I've been trying to wrap my head around the what, why and how of $\lambda$-calculus but I'm unable to come to grips with "why does it work"?

"Intuitively" I get the computability model of Turing Machines (TM). But this $\lambda$-abstraction just leaves me confounded. 

Let's assume, TMs don't exist - then how can one be "intuitively" convinced about $\lambda$-calculus's ability to capture this notion of computability. How does having a bunch of functions for everything and their composobility imply computability? What am I missing here? I read Alonzo Church's paper on that but I'm still confused and looking for a more "dummed down" understanding of the same.
You're in good company. Kurt Gödel criticized $\lambda$-calculus (as well as his own theory of general recursive functions) as not being a satisfactory notion of computability on the grounds that it is not intuitive, or that it does not sufficiently explain what is going on. In contrast, he found Turing's analysis of computability and the ensuing notion of machine totally convincing. So, don't worry.

On the other hand, to get some idea on how a model of computability works, it's best to write some programs in it. But you do not have to do it in pure $\lambda$-calculus, although it's fun (in the same sort of way that firewalking is). You can use a modern descendant of $\lambda$-calculus, such as Haskell.