The acceleration is though to the the result of a constant force which is somehow built-in to space - each volume of empty space contains something that results in this force. There are various ideas, mostly to do with some quantum effect, but no-one really has any idea what it is, and so it's given the pretty meaningless name 'dark energy'.
What will the future be? Dark energy seems to stay constant in a given volume of space, so it won't ever grow large enough to rip galaxies, planetary systems, stars, or even people, apart. All these things will be held together by the vastly stronger forces of gravity, electromagnetism, and nuclear forces. But on larger scales the dark energy force will overcome gravity. In many billions of years from now distant galaxy clusters will have accelerated away beyond our ability to see them. The universe will appear much, much smaller.