The concept you're referring to is related to the phenomenon known as gravitational time dilation, which arises from the theory of general relativity. According to this theory, the presence of mass and energy curves spacetime, causing it to be distorted. In regions where the curvature of spacetime is more pronounced, time itself flows at a slower rate relative to regions where the curvature is weaker.
When we say that matter is attracted in the direction where the rate of time is slower, it means that objects are influenced by the curvature of spacetime caused by the presence of mass. The object follows a curved path in spacetime, and this curvature results in what we observe as gravitational attraction.
It's important to note that this phenomenon is not caused by any specific force acting between the intervals themselves. Instead, it's the curvature of spacetime itself that guides the motion of objects. The presence of mass causes a curvature in the fabric of spacetime, and objects move along the paths determined by this curvature.
In a sense, you can think of the curvature of spacetime as the "medium" through which objects move. Objects follow the paths dictated by the geometry of spacetime, and this geometry is influenced by the distribution of mass and energy.
So, there isn't a physical "thing" between the intervals that moves one object towards the other. Instead, the curvature of spacetime, caused by the distribution of mass and energy, determines the motion of objects and gives rise to the phenomenon of gravitational attraction.