Бархатные тяги [barhatnye tyagi] literally means “velvet pulls,” but in this song it is playful slang for stylish, eye-catching shoes or sneakers. The unusual combination of luxurious бархатные [barhatnye] (“velvet”) with slangy тяги [tyagi] (“kicks”) gives the song its memorable fashion-forward swagger.
This is a pure dress-up-and-go-out party track from Gazan. The hook says it all: "А мы стиляги, на нас такие бархатные тяги" (and we're the stylish ones, look at the velvet kicks on us). Стиляги were the flashy, fashion-mad kids of the Soviet 1950s, and today the word just means people who dress to be seen. Тяги is youth slang for sneakers, so the whole flex is about shoes. They are also "мамины симпотяги" (mama's cuties), which is a cheeky, self-mocking way of calling yourselves handsome.
The night itself is gloriously small-town. They lace up tight, head "вечером на танцули" (to the dancing in the evening, танцули being a playful word for a dance party), and stop for a чебурек on the way, a greasy fried meat pastry you buy from a kiosk. Everyone in the neighborhood knows them ("на районе знает каждая собака", literally every dog in the area knows). They're broke, "даже если мы на нуле" (even if we're at zero), but still "на коне" (on a horse, meaning on top of the world). My favorite line is the most ridiculous one: "На посаженной Приоре обгоним Порше" (in a lowered Priora we'll overtake a Porsche). The Lada Priora is a cheap Russian sedan, and посаженная means its suspension has been dropped to look cool. Beating a Porsche in it is a joke, and the song knows it.
Compare that to "круче чем фаберже" (cooler than Fabergé, the famous jeweled eggs) and you get the attitude: no money, total swagger, great shoes.