There are two source papers by Fraïssé, a brief note and a longer text:

The abstract of the 1954 paper summarises Fraïssé's motivation, to generalize the ordering of the rationals:

This article is an elementary study of certain classes of relations,
$\Gamma$ and $\gamma$, which generalize the class of orders and that
of finite orders. We generalize the fact that any set can be ordered,
by generalizing the following two propositions: given two total
orders A, B, there exists an extended order (e.g. A + B) common to A
and B or their isomorphs; given a set of orders A, there exists an
extended order R common to A (or their isomorphs). In this way we
generalize two propositions on the order $\eta$ of the rational
numbers: any countable order is isomorphic to a restriction of $\eta$;
Order $\eta$ is the only countable dense order with no first or last
element.

For an English text, see volume 2, chapter 11 of Fraïssé's Course of Mathematical Logic.