Radar beam height is the altitude above the ground at which a radar's beam is sampling the atmosphere at a given range and elevation angle, taking Earth curvature and the standard bending of microwaves through the atmosphere into account.
Why the beam climbs with range
A radar antenna tilts up by a small angle when it scans, and Earth's surface curves away underneath the beam. Both effects combine so that even the lowest operational tilt on the WSR-88D, about 0.5 degrees, is a couple of hundred feet up right next to the radar and climbs thousands of feet above the surface at long ranges. At 100 nautical miles that same 0.5 degree tilt is sampling near the middle troposphere, well above where surface weather is happening.
The microwave beam also bends slightly through the atmosphere because the refractive index decreases with height. The standard-atmosphere assumption approximates that bending by treating the beam as traveling in a straight line above an Earth of four-thirds its true radius, which is where the familiar beam-height curves come from.
How forecasters use it
The first use is calibration of intuition. A hook echo 90 nautical miles from the radar and a hook echo 15 nautical miles from the radar are being sampled at very different altitudes, and the more distant image is looking at a level well above where a tornado might be. Every warning meteorologist keeps beam height in the back of their mind when reading structure and velocity on any given cell.
The second use is choosing the right tilt. When a nearby storm needs a look at the low levels, the operator drops to the lowest tilt available. When a distant storm needs a check on the mid-level updraft, a higher tilt at the same range may sample the height of interest more directly than the 0.5 degree beam is. The three-dimensional structure of a volume scan is really a stack of curved slices, each at its own height, and beam-height reasoning is what turns that stack into a physical picture of the storm.
Where beam height misleads
The standard-atmosphere assumption is convenient but not always correct. Strong low-level temperature inversions can trap the beam and bend it downward, causing anomalous propagation returns from the surface that appear at ranges where the nominal beam would be well above the ground. Elevated cool layers can cause the opposite effect and push the beam higher than the chart says.
Beam height also does not describe what fraction of the beam is filled. A tornado with a diameter much smaller than the beam width is barely sampled even when the reported beam altitude is low, because the vortex fills only a small portion of the pulse volume and its true winds get averaged with the calmer surrounding flow. Reading radar honestly at long range means combining beam height with beam width, with the knowledge that Earth curvature and beam broadening both grow with range, and with the awareness that the lowest tilt is not the same thing as the surface.