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This is oversimplified and I'm using common analogies but maybe try this: think of a mass floating in space and that mass distorts "space-time" around it - like a bowling ball sitting on a rubber sheet but in 3 dimensions not just a two dimensional surface. As you approach that area of "space-time" the local geometry is distorted so a "straight line" path to the traveller will actually be a curved path to an outside observer since the path follows this distortion in space ...errr space-time. Moving along that curved path produces forces on the moving object, similar to when you make any turn and feel acceleration forces pushing you toward the center of curvature. That force you feel is "gravity". As you get closer to the object causing the distortion, the curvature is greater and gravity forces are greater.
I'm no expert by any means, but I'm using "space-time" instead of just "space" since my limited understandinig is that space-time is a more accurate way for us to think of the "space" in our universe.
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