Spermatogenesis starts at the age of puberty due to significant increase in the secretion of gonadotropin releasing hormone $(GnRH)$ . This, if you recall, is a hypothalamic hormone.

The increased levels of $GnRH$ then acts at the anterior pituitary gland and stimulates secretion of two gonadotropins -luteinising hormone $(LH)$ and follicle stimulating hormone $(FSH)$ .

$LH$ acts at the Leydig cells and stimulates synthesis and secretion of androgens. Androgens, in turn, stimulate the process of spermatogenesis. $FSH$ acts on the Sertoli cells and stimulates secretion of some factors which help in the process of spermiogenesis