Euclid didn't define "line" by an axiom; in one translation, he defined it as "breadthless length," which is the kind of definition that only helps you if you already know what a line is. Isaac has it right when he suggests that what's important about a line is what it does, not what it is. An alternative approach is to found Euclidean geometry on analytic geometry. Define the real number line, then the Cartesian coordinate system, then define a line to be the set of solutions of $ax+by+c=0$ for fixed $a,b,c$ with $a$ and $b$ not both zero.
You asked for something that wasn't a proof or formal argument, so I hope this helps.
In any geometry, including non-Euclidean geometry (e.g. hyperbolic, or spherical geometry), "straight lines" are really called geodesics, which are defined to be the shortest line between two points. This means you stand somewhere holding one end of some rope, your friend stands somewhere else holding the other end, and together you pull the rope taut and this gives you your shortest path.
For example, say you're standing on the surface of a ball (i.e. a 2-sphere, such as the surface of the Earth), and your friend is some way away also on the surface, both of you holding the rope tight. This is spherical geometry. The taut rope or our "straight line" or geodesic is really the shortest path between us that lies on the surface, i.e. where the rope goes. This geodesic will look curved to someone in Euclidean space because there, the geodesic/"straight line" would pass through the ball.
Therefore it turns out that our definition of "straight" depends on the geometry we're using and how we pull the rope taut (the metric we use). It just so happens that in Euclidean geometry, this gives us lines that we call straight.
Interesting note: in other spaces, there are some super cool and peculiar metrics that make the taut rope (shortest path) go into weird shapes in Euclidean geometry! One example: https://en.wikipedia.org/wiki/Taxicab_geometry
Best Answer
In modern standards those definitions don't make any sense other than just an intuitive picture of what is intended. And actually even in this case it's not very clear at first what Euclid refers to with that definition of "straight line" if you don't already know what a straight line is.
Today, when you want to study something, you start with objects, which are said to be undefined. They are, well, just things that satisfy certain axioms. In the case of Geometry, you may consider straight lines to be undefined or maybe you can say that the undefined objects are points and that lines are set of points which satisfy certain axioms, which depend on the book you're reading.
Also, as already pointed in the the comments, Euclid's Elements is a book with lots of "holes" that need to be filled by the specialist. In that case you can say that Euclid's elements is more like a sketch. Knowing this, the easy way to read it is just going through it informally by trusting your intuition and using the postulates, just like he did. Along the way you can takes notes about the facts that bugs you the most and try to guess what is missing.
Once you understand what Euclid's Elements is about, then you can skim very quickly how David Hilbert filled those holes of Euclid's in his "Foundations of Geometry" or if you like you can Greenberg's book "Euclidean and no-Euclidean Geometry".