Why are computable functions also called recursive functions?
The founders of computability theory were mathematicians.
They founded what is now called computability theory 
before there was any computers. 
What was the way mathematicians defined functions that could be computed?
By recursive definitions!

So there were recursive function before 
there were any other model of computation like Turing machines or 
lambda calculus or register machines.
So people referred to these function as recursive functions.
The fact that they turned out to be exactly 
what Turing machines and other models can compute is a later event
(mostly proven by Kleene).

We have the simple definition of a recursive function which is now called
primitive recursive function.
There were not general enough (e.g. Ackermann's function) 
so people developed more general notions
like $\mu$-recursive functions and 
Herbrand-Gödel general recursive functions that
did capture all computable functions (assuming the Church's thesis).
Church claimed that his model of lambda calculus captured 
all computable functions.
Many people, and in particular Gödel, were not convinced that
these capture all functions that can be computed.
Until Turing's analysis of computation and introduction of his machine model.

The name of the field used to recursion theory.
However there has been a successful push in recent decades 
to change the name to something more appealing from
recursion theory to something more computer sciency (vs. mathy). 
As a result the field is now called computability theory.
However if you look at books, papers, conferences, etc. in the early decades
they are called recursion theory and not computability theory.
Even the title of Soare's own 1987 book 
(who was the main person behind the push to change the name 
to computability theory) 
is "Recursively Enumerable Sets and Degrees".

If you want to know more about the history 
a fun and good place to read about it is 
the first chapter of Classical Recursion Theory by Odifreddi.