Let's say I'm in empty space, far away from everything and I throw a ball forward at a constant speed. The ball will move in a straight line away from me. (I don't think anybody can disagree with this)
Not only is this a straight line in space but it is also a straight line in spacetime. (Why it is also a straight line in spacetime takes a while to explain, so I'll skip it for now unless you're interested.)
Now let's say I'm near the Earth. The Earth warps the nearby spacetime into a new shape. I throw the ball again, this time it falls toward Earth. It has again followed a straight line in spacetime. However because the space has a different shape, straight lines usually lead to Earth.
This is a shape called a monkey saddle, with straight lines in the space drawn in it. As you can see straight lines in a curved space can be very different from straight lines in a flat space.
Gravity is nothing more than particles following straight lines in spacetime. This is why two objects of different masses fall at the same speed, following a straight line doesn't depend on your weight.