What is motivation behind the definition of a complete metric space?
Intuitively,a complete metric is complete if they are no points missing from it.
How does the definition of completeness (in terms of convergence of cauchy sequences) show that?
[Math] Motivation behind the definition of complete metric space
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To show that a closed interval is complete, you can do the following: You first show that every Cauchy-sequence is bounded. It therefore has a convergent subsequence and the limit is in the interval since it is closed. Then you verify that if a sub-sequence of a Cauchy sequence converges to a point, the sequence itself converges to the same point.
Metric spaces were introduced by Frechet in his PhD dissertation on functional analysis, in 1906. Functional analysis (and rigorous modern analysis) was still quite new at the time. Also, an abstract, axiomatic, approach to mathematics was also not yet as routine as it is for us today. The mathematicians of that time were studying various spaces (mainly spaces of functions) and they had various notions of convergence in such spaces. But everything was ad-hoc. For each space its own notion(s) of convergence was introduced, and studied. Of course, similarities were noticed (i.e., uniqueness of limits was a common trend etc.). There was a dire need to simplify things and unify arguments. Frechet's genius was to do just that by axiomatizing the notion of distance and show that many of these spaces were instances of metric spaces. Then, by proving one result axiomatically from the metric axioms, it automatically holds for all instances.
This was the historical motivation. In the modern view, the concept of a metric space is just an axiomatization of the notion of distance. It is among the more straightforward axiomatizations, especially to modern students who see axiomatic systems early on. The notion of distance is very important since, for instance, it is used in the definition of limit. Many geometric notions rely on a notion of distance (e.g., circles). So, it is natural to distill some common properties of distances in various contexts and set them as axioms. Voila - metric spaces.
Just as a side note: There is quite a lot of flexibility in the axioms of metric space. Neglecting any of them (giving rise to things like semimetric, quasimetric spaces etc.) give interesting spaces as well with somewhat similar theory as metric spaces exhibit. One crucial difference though is that if the symmetry axiom is neglected (quasimetric spaces then), then the general theory is quite difference (in particular, there are then many different completions).
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I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition.
A somewhat related notion of completeness is the geodesical one. The definition may not be too much appealing unless you're interested in differential geometry, but one of its consequences is easy to explain: if a Riemann manifold is geodesically complete, you can join any two points by a length minimizing geodesic. (But geodesic already implies that it minimizes length, doesn't it? Not quite: just locally. So, for instance, the meridian joining the North Pole with London, but going "backward", through the Bering Strait and the Pacific Ocean, then the South Pole, Africa and finally London, is a geodesic, but not a length minimizing one blatantly.)
Anyway, $\mathbb{R^2} \backslash \left\{ (0,0)\right\} $ is not geodesically complete, since there is no length minimizing geodesic joining, say, $(-1,0)$ and $(1,0)$, due to the "hole" $(0,0)$. At the same time, as a metric space, $\mathbb{R^2} \backslash \left\{ (0,0)\right\}$ is not complete: the Cauchy sequence $(\frac{1}{n}, 0)$ converges to $(0,0)$, but since $(0,0)$ is not in $\mathbb{R^2} \backslash \left\{ (0,0)\right\}$ it doesn't have a limit there.
Well, the Hopf-Rinow theorem tells us that this kind of things always happen together: a "hole" for geodesics is the same as a "hole" for Cauchy sequences, since for a (finite-dimensional) Riemann manifold $M$, both notions agree: $M$ is complete as a metric space if and only if it is geodesically complete.