On a (digital) TV, sharpness controls a peaking filter that enhances edges. That is not so useful on a display if used as a computer monitor.
In the previous century, on a high-end analog CRT monitor, sharpness may have controlled the focus voltage of the electron gun. This affects the spot size with which the picture is drawn. Set the spot size too small (too sharp) and the line structure becomes too visible. Also there may be annoying "Moiré" interference with the structure of the shadow mask. The optimum setting depends on the resolution (sample rate) of the picture, as many CRT monitors were capable of multiple resolutions without scaling (multi-sync). Set it just sharp enough.
High-end CRT TVs had Scan Velocity Modulation, where the scanning beam is slowed down around a vertical edge, and also a horizontal and vertical peaking filters and perhaps a horizontal transient improvement circuit. Sharpness may have controlled any or all.
Sharpening in general enhances edges by making the dark side of the edge darker, the bright side brighter, and the middle of the edge steeper. A typical peaking filter calculates a 2nd order differential, in digital processing e.g. (-1,2,-1). Add a small amount of this peaking to the input signal. If you clip off the overshoots then it reduces to "transient improvement".
On some digital devices, the sharpness of a scaler may be controlled, e.g. in my digital satellite TV receivers. This sets the bandwidth of the polyphase filters of a scaler, which converts from a source resolution to the display resolution. Scaling cannot be perfect, it is always a compromise between artefacts and sharpness. Set it too sharp and annoying contouring and aliasing are visible.
This may be the most plausible answer to your question, but only if the monitor is scaling. It would do nothing for an unscaled 1:1 mode.
Source: 31 years of experience in signal processing for TV.