The Levi-Civita field's main claim to fame is that it's the smallest real-closed field which is a proper extension of the reals and which is Cauchy-complete. That last bit is important, or else the Puiseux series would instead take the crown: it is the real closure of the field of formal Laurent series on the reals.
My main question is why this is an important quality to have, or how to make intuitive sense of the particular definition of Cauchy completeness.
The reals satisfy a notion of completeness in that every monotonic and bounded sequence of reals converges to a real number, and in fact they are the unique ordered field that does so. For instance, in any ordered field except for the reals, the sequence $(1, 1/2, 1/3, ..., 1/n, ...)$ is bounded, but won't converge to any unique limit, as there will be many infinitesimals smaller than each term in the sequence but no "largest infinitesimal." Similarly, the sequence $(3, 3.1, 3.14, 3.141, 3.1415, ...)$ doesn't converge to $\pi$ in any ordered field except for the reals. But our notion of Cauchy completeness relaxes this so that rather than each sequence being "bounded," it instead needs the elements to eventually get "arbitrarily close together" (closer than any positive element of the ordered field). Thus, these sequences no longer serve as counterexamples to the completeness of our ordered field, because they no longer satisfy the definition of "Cauchy sequence" unless we are in the reals to begin with: otherwise, there will be infinitesimals which are smaller than the difference between any pair of elements in the sequence.
I basically just would like to know why this is an important property to have. Although I'm sure my intuition is wrong, it seems to me as though this definition of completeness is somewhat contrived: you've excluded all of the important examples that would rule out non-Archimedean fields based on what feels like a technicality, but it isn't like you've really generalized things to look at ordinal length sequences, or ditched sequences entirely and are looking at sets, etc. Why is this particular definition important and what can you do in the Levi-Civita field that you can't do with Puiseux series?