Think of your velocity vector as having a time component. The magnitude of this vector is c, so when you are at rest, you're moving full speed through time. When you accelerate, you shift some of this speed into the spatial dimensions. This is also why time passes more slowly for moving objects. Gravity also has this effect because not only is space curved, but space-time is curved. This means what would normally be a straight path through time is partially warped into the spatial dimensions when you encounter such a curvature.