Let us continue analysing braking. Weight transfer during accelerating and cornering are mere variations on the theme. We won't consider subtleties such as suspension and tyre deflection yet. These effects are very important, but secondary. The figure shows a car and the forces on it during a "one g" braking manoeuvre. One g means that the total braking force equals the weight of the car, say, in pounds.
In this figure, the black and white "pie plate" in the centre is the CG.
G is the force of gravity that pulls the car toward the centre of the Earth. This is the weight of the car; weight is just another word for the force of gravity. It is a fact of Nature, only fully explained by Albert Einstein, that gravitational forces act through the CG of an object, just like inertia. This fact can be explained at deeper levels, but such an explanation would take us too far off the subject of weight transfer.
Lf is the lift force exerted by the ground on the front tyre, and Lr is the lift force on the rear tyre. These lift forces are as real as the ones that keep an airplane in the air, and they keep the car from falling through the ground to the centre of the Earth.
We don't often notice the forces that the ground exerts on objects because they are so ordinary, but they are at the essence of car dynamics. The reason is that the magnitude of these forces determine the ability of a tyre to stick, and imbalances between the front and rear lift forces account for understeer and oversteer. The figure only shows forces on the car, not forces on the ground and the CG of the Earth. Newton's third law requires that these equal and opposite forces exist, but we are only concerned about how the ground and the Earth's gravity affect the car.
If the car were standing still or coasting, and its weight distribution were 50-50, then Lf would be the same as Lr. It is always the case that Lf plus Lr equals G, the weight of the car. Why? Because of Newton's first law. The car is not changing its motion in the vertical direction, at least as long as it doesn't get airborne, so the total sum of all forces in the vertical direction must be zero. G points down and counteracts the sum of Lf and Lr, which point up.