The best analogy I have ever heard to try and visualize this is to imagine a sheet of cloth pulled taught so that it is a flat sheet suspended in the air. Then, drop a bowling ball in the middle. The sheet will bend around the bowling ball, making a pretty massive indentation in the sheet. The further you get from the ball, the less indentation there is, but the sheet is _always_ intended a little bit no matter how far away you get.
That is gravity. Objects with large mass (the bowling ball) create large indentations in space-time (the sheet). The closer you are to the object with mass, the greater the effect of gravity (the indentation) but gravity is always there, no matter how far away you get.
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