He is explaining that he wears Gucci (the video clearly showcases Gucci clothing during this lyric) clothing, which also matches the interior of his car.
i have 20" diameter rims/wheels on my vehicle, and since i call myself Birdman i will express my pride by claiming to pop my feather - the human equivalent to a bird preening itself.
A whip is a commonly used word for your car, if you have an "immaculate whip," you have a sick car. New shoes for your whip, means you got new tires, or new rims, or both.
expensive aftermarket tires are needed for this vehicle, and they have to be the run-flat type of tires that when punctured allow you to continue driving. this implies not wanting to change your own tire or wait for a tow truck, or having to evade police.