Can I ride faster with a bike jersey?
No, you will not ride faster in any meaningful way unless you're doing time trials at an elite level where mere seconds (or less) of improvement are gained through optimizing a long list of equipment (with clothing in the middle of that list). As always, the overwhelming determinant of performance is training. 

The real reason for wearing a jersey is the following:


Sweat management. Wicking fabric has to lie against to the skin to actually wick the sweat and transport it to the exterior where the wind will evaporate it quickly.
Pockets in rear. Pockets are useful. The only place you can reasonably put them is in back.
Long rear. It looks better when you're hunched over the bike.
A zipper. So you can regulate temperature.
Fitted + Flat seams. Flapping fabric is annoying on a ride and seams tend to painfully rub on long rides.