"Real cause of gravity"

gravity caused by curved space - am I right?

I was told that gravity is caused by warped space time (according to, is it GR or SR). I was told that when you move into a curve, you accelerate. This is what was really confusing me. I tried to picture something moving faster because it was going from a straigh path to a curved one but I just wasn't seeing it. How does moving along a curve translate to increasing speed? Then it dawned on me that one thing I could see was that if the curve was getting tighter and tighter, as 'twere, one thing that does increase and that is the rate at which the object in question is changing directions! So the picture I formed in my head is this: Imagine the earth by itself in space (to avoid distractions). Now pop something into existence, say, a person, some ways away from the Earth. Now this person, having just popped into existence, instead of just floating in space will start to move towards the earth, slowly at first and then faster and faster, making a straight path. Now, this straight line being traced is what we see, but the reason the person is accelerating is because this is not the whole picture. What we see as a straight line path is actually a person changing directions because they are actually travelling in a curve and the reason they are accelerating is because the curve they are caught on is getting more and more curved. In other words, the faster speed we see is the increased rate the person is changing direction on the ever tightening curve.