Because the sensor essentially never stops sensing. There is no mechanism built into the sensor to not bleed charge from the little capacitors when light hits it. It also takes long enough to read out all the data from the sensor, so that there would be significant exposure time variation between parts of the image if the firmware reset the sensor, waited for the exposure time, then read out the data.
Digital sensors can read successive frames of video without needing mechanical shutters. However, the frame rate is fixed and known, the exposure time per frame is quite long (relative to typical still photography values), "rolling" exposure where different parts of the frame are exposed at different times is acceptable, and resolution is low.
Some sensors do essentially have "electronic shutters". These can integrate the light for a specified time, then freeze the value at least long enough for the whole array to be read out. However, this takes silicon area, which is usually more desirable to spend on better light gathering in digital still cameras.
To get the accurate and fast shutter times and the high resolution expected of today's digital cameras requires a mechanical shutter with current technology at prices that the market will accept.