Remember that the anterior pituitary (adenohypophysis) originated from the upper pharynx and is not directly connected to the hypothalamus at all. The hypothalamus is still able to control the activity of the adenohypophysis but through different means as with the neurohypophysis. Control over the adenohypophysis is exerted through what's called a portal system; the specific name for this portal system is the hypothalamohypophyseal portal system (say that 10 times fast). Remember a portal system is two consecutive capillary beds that are joined together by a portal vein.
Normally as blood flows through a capillary bed there are only one set of capillaries within that bed. On one side of a normal capillary bed there is an artery (metarteriole to be specific) and a draining vein (venule). So blood flows right in and right out of the one set of capillaries. With a portal system blood is circulated into the first capillary bed by an artery and is usually called the primary capillary bed. Instead of the blood flowing directly into the venous circulation it instead flows through a portal vein and into a second capillary bed, called the secondary capillary plexus. Connected to the secondary plexus is a venule for drainage of blood into the venous circulation. This is a good way of transporting materials directly from one tissue to another without having to travel the entire length of the circulatory system first.
The primary capillary plexus is found on the infundibulum (stalk) of the pituitary gland and is formed by the superior hypophyseal arter; which is a branch of the internal carotid artery. The drainage vein of the secondary plexus is called the hypophysial vein and drains directly into the dural sinuses which drain into the venus circulation.
The hypothalamus secretes what are called releasing and inhibiting hormones. Both of these are secreted into the primary capillary plexus from various centers of the hypothalamus. Both of these also travel in this order: primary capillary plexus → portal vein → secondary capillary plexus → diffuse into adenohypophysis. There are specific cells in the adenohypophysis (will be covered and named later) that synthesize and secrete their specific hormones. Releasing and inhibiting hormones are very specific to their cell type. For example, thyrotropin releasing hormone (TRH) only stimulates thyrotropes (cells that produce the hormone TSH) to secrete its hormone. You can judge by the name that releasing hormones increase that manufacturing and secreting activities of these cells while inhibiting hormones do the opposite.
Source: Mind of Aaron
This video covers how the hypothalamus controls the activity of the anterior pituitary (adenohypophysis).
Source: Self made