How does a 44E21D hall sensor work? And how to wire?
The chip looks like a hall effect sensor.
As the IR remote receivers in that package there is not only the sensor but also the appropriate circuitry to drive and read it. Two pins are probably power (\$V_{CC}\$ and ground) while the third is the output.
Looking at your PCB traces I'd make this guess:


pin 1: \$V_{CC}\$
pin 2: ground
pin 3: output


And the guess agrees with Handoko's link. You can confirm this by powering the board and checking the voltage across pin 1 and 2, or following the traces and seeing where they connect to other components (that's what I did).
Pin 2 connects to a huge copper plane, these planes are usually connected to ground to provide shielding and to carry around the reference voltage.
Pin 1 is connected to a trace that wanders a lot, goes to a couple of connectors and to the microcontroller: no way that's the signal output, that must be \$V_{CC}\$.
Pin 3 finally seems to connect directly (and only) to a microcontroller pin: now that's your output. If you look closely you can even see the resistor shown in Handoko's link.  

My pin order: refer to the last photo, pin 1 is at the bottom while pin 3 is the top one. Pin 2 is in the middle of course.

Addendum: to test it you do not need your motor, a hall effect sensor senses magnets: a fridge magnet or something similar is probably quite enough for it to work. You will need a multimeter or something to tell if the output is high or low. Please keep in mind that the sensor output might not be strong enough to directly drive a LED.