In a 3-D sense, a beam of radar emission has a cone shape. The tip of the cone is at the radar site and the diameter of the cone gradually increases away from the radar site. As a radar beam moves further from the radar, it expands to take up a larger volume. An example of this process is to turn out the lights and shine a flashlight on the wall. First stand 2 feet from the wall and then gradually stand back further from the wall. You will notice as you keep standing further from the wall that the diameter of the brightest light striking the wall becoming larger and larger. You will also notice some light from the flashlight does not stay in the main flashlight beam. A small portion of the light fills the room (you will notice that when you shine a flashlight on the wall that the light becomes dimmer the further away it is from the bright spot on the wall). The light that is not in the main beam is similar to "side-lobes" that are produced by radar.
A process called attenuation occurs as a radar beam of radiation leaves the radar site. Substances in the air absorb some radar radiation including hydrometeors. The more space a beam travels through, the more absorption that will take place of that beam. Energy backscattered from a storm near the radar will be more powerful than the energy backscattered from a distant storm. Backscattered radiation close to the radar does not undergo much attenuation. This causes rainfall intensity to be measured higher for storms near the radar as compared to far from the radar, all else being equal.
The resolution of radar data decreases with distance from the radar. A "range gate" is the area encompassed by one pixel of radar data. These range gates become larger with increasing distance from the radar site since the beam is expanding as it moves away from the radar site. Large range gates will result in less resolution. An analogy is to think of two computer images where one looks crisp and sharp while another looks grainy. The grainy image has larger pixels and thus has a poorer resolution. The consequence of having a larger range gate results in an overestimation of the areal extent of rainfall. The value of dBZ in a range gate represents the average dBZ within that range gate. Also, the entire range gate must be filled with a dBZ value. In actuality, precipitation and/or heavy precipitation may only be occurring in part of the range gate. Since every range gate must be filled that has reflectivity coming from it, this causes the area that precipitation covers to be larger on radar than in actuality. This error increases with distance from the radar.
In situations where only part of the range gate is filled, the reflectivity will be lower in that range gate than is occurring in actuality with the precipitation that is occurring in part of the range gate. The reflectivity will be lower because the reflectivity must be averaged over the entire range gate. As range gate size increases, it becomes increasingly more likely that only part of the range gate is being filled with precipitation and/or heavy precipitation.
To sum up, as beam spreading increases (diameter of cone increases), rainfall intensity is increasingly underestimated, rainfall areal coverage is increasingly overestimated and precipitation and/or heavy precipitation in only part of the range gate increasingly becomes averaged over a larger area.
The image below shows the errors produced by beam spreading. The radar site is in Columbus. Notice as you go further south the range gates become larger. The storms at the edge of the radar look more grainy and larger than the storms close to the radar. The storms near the edge of the radar are in actuality smaller than shown by radar. The storms near the edge of the radar are also being sampled at a higher elevation than the storms close to radar.