Reading output from "line-driver" hall-effect sensor
A "raw" Hall effect sensor will give an output voltage proportional to the current and orthagonal magnetic field.



Figure 1. The principle of the Hall effect. Source: Wikipedia Hall effect.


In your case it seems as though you have two output terminals per sensor. These would correspond to \$ V_H+ \$ and \$ V_H- \$ in Figure 1. 
Since the Hall effect produces quite low voltages it seems reasonable to assume that some amplification of the signal takes place in the chip.
Since the chip has single rail supply it also makes sense that the voltage would vary around mid-supply. This is 5.8 V in your case.
It also seems plausible that there are three possible states for the sensor:


North pole at sensor.
Neither or balanced / cancelling / opposing poles at sensor.
South pole at sensor.

These three options would explain your three output voltages. More than likely you should find that when A+ is high that A- is low and vice versa. You should also find that when A+ is neutral that A- is too.


To "decode" these signals you will need three analog comparators.



Figure 2. The only similar circuit found on a web search shows the three sensors in series rather than parallel. This is an efficient approach where possible as the series current reduces overall current consumption. The TLC3704 contains four comparitors. Image source: Flickr.