Gravity pulls down on the plane itself plus all of the people and baggage inside, and every single air molecule that is shot through the engine or collides with the fuselage or wings pushes on the plane as well
1. Curved shape of the wings and their slightly inclined angle means that the bottoms smash into more air molecules than before (and smash harder into those molecules), so the pressure on the bottom of the wing goes up
2. Fewer air molecules now strike the top of the wing and those that do strike it less forcefully, partly because it's being "shielded" by its own forward motion and partly because a curving stream of air has lower pressure on the inside of the curve since the molecules get thrown centripetally to the outside
Even in inefficient engines, the spinning fan blades get their horizontal lift, which we call "thrust," by moving quickly through air with a curved shape and a slightly inclined angle – they're essentially mini-wings
An airplane is essentially a meta-wing: it flies by moving mini-wings fast enough to push air molecules backwards, which moves the plane forward fast enough that its big wings push air molecules down