Water level rising in the brine tank pushes the float up. It shouldn't stick in the up position so either there is interference, mineral buildup or a defective part causing it to stick in the up position. You should investigate to see if you can determine the reason for sticking.
The float level might be misadjusted--it should be adjusted to cut off the water supply just before an overflow/spill would occur. If it is adjusted correctly you should confirm there isn't a build up of crud in the bottom of the brine tanke that is elevating the water level. If that doesn't solve the problem there is probably a problem with the softener valve that is allowing to much water into the brine tank. The quantity of water is controlled by a Brine Line Flow Control (BLFC) and the timer. If the BLFC is damaged or missing that could cause the problem. As timer that is not set correctly or not functioning correctly could cause the problem. There could also be a problem with the valve itself--could be the brine valve or could be the main valve. You didn't tell us the valve manufacture so it is difficult to be more specific about what /how to check out the softener valve.