Constant acceleration simulates gravity because what we call gravity is constant acceleration of an object towards the centre of an idealised mass. We stick to the outside of the earth because we're constantly being accelerated towards its centre; the surface of the earth pushes back. Acceleration, or a change in velocity, is measured in terms of "gravities" or G, which is where the term "g-force" comes from; if an object is accelerated in one direction, it experiences a "gravity" effect in the opposite direction. That's why you feel yourself pressed into a the seat of your car when you accelerate. If you had no external reference, you would believe you were lying on your back or that the car had suddenly pointed up into the air. This is the reference frame, one of the basic concepts of relativity.
If you are inside a box with no windows, reducing your reference frame to what you can see and experience inside that box, and feel an apparent gravitational force, you cannot tell if you are being pulled down by a large mass, or if the box is accelerating upwards, because in both cases the effect is the same; you feel a "force" pulling you to the floor of the box. Give the box a window and you will be able to tell which is which, but you'll still feel the same apparent "force". A person standing inside a spaceship accelerating at 1G (that means it's constantly going faster) will experience earth-equivalent gravity until the ship stops accelerating.