What seems to be missed on these answers is our motion through spacetime. Spacetime is the "web" of both space and time. Our galaxy is hurdling through the universe at tens/hundreds of thousands of mph as we are also hurdling through time, - and again space and time are part the same "matrix". So we are constantly hurdling through spacetime, and that is why when mass compresses and curves spacetime, we have a tendency to "move" toward mass. There is no such thing as a force of gravity, it is not a force, it is an observed effect. We feel as if we are constantly being pushed to the ground because of us flying through space and time so fast, and that spacetime that we are flying through is curved toward the mass of the earth. Actually, the earth is pushing, and exerting force on us.
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Please start with the Wikipedia article Introduction to General Relativity. Once you've gone through that, you will probably more specific questions, and you'll be more familiar with the terminology you need to both ask them and to understand the answers.
Still don't quite understand exactly why do things fall down, why are they attracted towards each other. I know that every mass causes the warping of spacetime around it, but how does then that curved spacetime attract other objects towards itself.
Objects travel in spacetime. "Free fall" means that the object is following a minimum energy path in spacetime. In flat spacetime, that minimum path is a straight line (changing direction requires a force, which means putting energy into the object).In curved spacetime, the minimum energy path is a geodesic, and generially is a curve (which is why it's called "curved spacetime").One of the "rules" of general relativity (and if you are not willing to invest the time to learn the subject, you won't be able to understand "why") is that those geodesics curve toward masses, not away from them. Hence, objects fall toward Earth, not up into the air.
Googling 'How does warping of space-time cause gravity' yields a lot of good answers.