The Star Wars: Incredible Cross-Sections factbook offers a host of reasons why the Empire favours TIE fighters. They're instantly recognisable, visibly imposing, impossibly fast, ultra-low maintenance, incredibly maneuverable and above all things, cheap cheap cheap to build.
Since the Empire evidently views their pilots as largely expendable, kitting them out with shields and heavy armour is a total waste of resources when they can sling them out of the ship (by the hundreds) in what amounts to little more than an engine with guns.
Hurtling through space, TIE fighters are the most visible image of the
Empire's wide-reaching power. The TIE fighter engine is the most
precisely manufactured propulsion system in the galaxy. Solar
ionization collects light energy and channels it through a reactor to
fire emissions from a high-pressure radioactive gas. The engine has no
moving parts, making it low-maintenance. To reduce the mass of the
ship. TIE fighters are built without defensive shields, hyperdrive
capability, and life support systems - so the pilots must wear
spacesuits. The light-weight ship gains speed and maneuverability at
the price of fragility and dependence on nearby Imperial bases or
larger craft for support.
There's also the fact that the ships fit in nicely with the Empire's martial philosophy of dehumanisation and sameness
TIE pilots may never use the same ship twice, and develop no
sentimental attachment to their craft as Rebels often do. TIE pilots
know that every reconditioned fighter is identical to a factory-fresh
ship: one is the same as many thousands - another reinforcement of
Imperial philosophy of absolute conformity.
On top of that, their lack of hyperdrive and life-support would presumably deter any thoughts of defection to the Rebellion.
As to why the Empire would field such a craft knowing of its obvious weaknesses, this is addressed in the Star Wars Databank article on TIE Fighters.
A TIE cockpit was cramped, and the fighter’s lack of defenses made flying one a dangerous calling. But TIE pilots took a perverse pride in the flaws of their craft. They saw the ability to fly a TIE effectively as the sign of true ability for a pilot, and TIE aces were held in great esteem by pilots who dreamed of amassing similarly impressive service records.
Star Wars: Databank - TIE Fighter