Generally the game of the opening is to develop the pieces (knights and bishops in particular) to active squares and to gain space for operation of the pieces (usually achieved by occupying the center with pawns).
If we are talking about the kings pawn opening, i.e. any opening that starts with 1. e4, there is little to say besides:
- white aims to occupy the center with a pawn on e4
- white makes space for the development of the light squared bishop on f1 (and potentially the queen)
Note that the game after 1.e4 could develop in all kinds of ways of very different character. Many of these openings have their own name (Sicilian, Ruy-Lopez, French, Caro Kann...), their own aims and objectives. Really I'd struggle to call a single move like 1. e4 an "opening".
Also often you will find that proper openings like those mentioned above will be listed by their name while the name King's Pawn Opening is reserved for all the rest, i.e. for games that don't follow any of the established openings.