VW Passat Won't Start (again). New Battery installed. Attached audio file of the sound it makes!

The first thing to do is verify whether the starter motor is working or not. A mechanic can check the voltages at both terminals of the starter motor during cranking (when the key is in “start”), from the terminal to the starter motor case. If either is below 9.6 volts during cranking, you have an electrical problem of some sort before the starter motor. That should prove relatively simple to fix. If both are above 10 volts (during cranking), and the starter motor isn’t turning, then the starter motor has to be removed and inspected. If the starter motor seems to be turning, but the engine isn’t turning, likewise, remove the starter motor for inspection. While the starter motor is removed, make sure the machanic rotates the flywheel through 360 degrees and checks the condition of all the gear teeth.