What makes lambda calculus relevant to study?
In computer science we want to analyze and understand source-code with mathematical rigour. That's the only way to prove interesting properties (such as termination) with absolute certainty. For that we need a language with a very well-defined meaning for every construct.

In theory this could be any language with a good formal semantics. But to make things less complicated and less prone to error, it's best to use a language that is as simple as possible but still able to express any program (i.e. is Turing complete). For reasoning about imperative code, there are Turing machines. But for reasoning about functional programming, there is the $\lambda$-calculus.

The basic $\lambda$-calculus is like a functional programming language, but with a lot of 'baggage' taken out. It's not important that this be a nice language to actually write programs in, nor that it be an efficient language. Just that it is simple and expressive. For example, we don't need loops, because we can simulate them with recursion. And we don't need functions with multiple parameters, since we can simulate them with Currying.

Now, at some point you may want to prove properties about constructs that are not part of the basic (untyped) $\lambda$-calculus. That's why computer scientists have extended it in different directions over the years. For example, to reason about type-systems there are a great many variations of typed $\lambda$-calculi.