We can add x and y by noting that x + y + (−(x+y)) = 0. We get −(x+y) by using the line thru x and y, and then use the above trick to get x+y = −(−(x+y)).
There is yet one more complication when two of arguments to L are the same. If you use calculus to compute the limiting line as two points approach each other; you get a formula for the tangent that can be computed using only rational arithmetic. This same formula works when we are using a finite field, even though calculus is out of its depth in that case.
I don’t know who discovered that this slightly obscure construction actually produces a commutative group. It is not obvious.
While there is no multiply there is yet more gold in introducing the notation “nx” where “n” denotes an ordinary integer and “x” a point on some cubic curve. Computing nx for large n is feasible thru the same trick that crypto code computes nm mod k for large m.
If we define our curve over some finite field all of the above magic remains intact.
If you are competent at algebra and have more fortitude than I you can make the above rigourous.