The short answer is: we do not know exactly, although we do have some insights.
I will take the example of puberty.
Although a clear definition of puberty is lacking, it is quite clear that it corresponds to a period where gonadal function starts.
This, in turn, is derived from the activation of the gonadotropin system, which consists of two main cell types:
a small number of neurons located in the preoptic area of the hypothalamus (a nucleus at the base of the brain) called the GnRH neurons. GnRH is the Gonadotropin-releasing hormone, a small peptide that stimulates the production of gonadotropins from the pituitary.
the gonadotrophs, a specialised group of cells in the pituitary (a gland located underneath the brain) which produce two hormones, called luteinizing hormone (LH) and follicle-stimulating hormone (FSH) which stimulate the gonads to produce various hormones, such as estrogen.
In mammals, the secretion of GnRH, and thus LH/FSH varies during the course of the menstrual/estral cycle. This cyclicity lasts several days (4-5 days in rodents, ~1 month in humans) and entrains the cyclical secretion of estradiol (E2) from the gonads.
Note that this is a sort of self-sustaining cycle, as cyclic levels of E2 will then allow for cyclic GnRH secretion and so on.
But, back to your question: how does the GnRH/LH/E2 system "wake up" at puberty?
The exact mechanism is still unknown, but recent work has found an important mediator, called kisspeptin, that is produced from two population of neurons in the hypothalamus, called the kisspeptin neurons.
Kisspeptin has been shown to be a very potent activator of GnRH neurons and work in the mouse has shown that these neurons appear at the time of puberty, their number increasing dramatically between 25 and 31 days (puberty is at around 30 days in mice).
Postnatal Development of Kisspeptin Neurons in Mouse Hypothalamus; Sexual Dimorphism and Projections to Gonadotropin-Releasing Hormone Neurons - Clarkson and Herbison - Endocrinology, 2006
Similar work exist in the monkey:
Increased hypothalamic GPR54 signaling: A potential mechanism for initiation of puberty in primates - Shahab et al. - PNAS, 2005
In humans mutations in either kisspeptin, or its cognate receptor GPR54 results in disturbances of pubertal maturation because of underactivation of the system (hypogonadotropic hypogonadism) or hyperactivation (precocious puberty).
So, now the question is shifted: why do kisspeptin neurons show up only at puberty? We don't know for sure, but it looks like increased levels of E2 could be important for this.
Again, we get into a self-sustaining cycle. Growth of the body generates an increase in E2 production (possibly due to increased volume of the gonads?), which, when over a certain level permits the development of kisspeptin neurons, which will then stimulate the GnRH neurons, resulting in increased LH and E2. We then have more E2 and this makes kisspeptin neuron grow even more etc etc.
From: Postnatal development of an estradiol-kisspeptin positive feedback mechanism implicated in puberty onset. - Clarkson et al. - Endocrinology, 2009