How are bones growing, if bones are not connected to the brain?
The developmental growth of bone tissue is hormonally controlled. It is, as far as I know, not under direct neuronal control. Before reaching adolescence, the long bones (mainly in the arms and legs) grow in the epiphyseal plate, the area of the bone where cartilage is formed and ossified on the diaphyseal side, thereby lengthening the bone. The longitudinal growth of long bones continues until early adulthood at which time the chondrocytes in the epiphyseal plate stop proliferating and the epiphyseal plate transforms into the epiphyseal line as bone replaces the cartilage (boundless website). The growth and stop of growth in adulthood is hormonally controlled. Bone tissue growth is controlled by various hormones including parathyroid hormone and growth hormone.

Growth hormone (GH) is secreted by the pituitary, in turn controlled by the hypothalamus in the brain. So although the brain is involved, it is not directly involved, in that it regulates growth, and cessation of it through hormone release. Growth hormone is probably the most important hormone and is released by the pituitary, but parathyroid hormone released by the thyroid, among other hormones, may play a role as well.   

Edit: Your question implies bone growth, which at first seemed to point toward developmental growth to me. However, there is also the more dynamic control of bone mass due to the forces that act on them in everyday life, referred to as bone mass accrual. This process is controlled by the sympathetic and parasympathetic system (Bajayo et al., 2012): 



This system acts through direct neural innervation of bone tissue and stimulates bone formation when needed (after imposed strain on the skeletal system), but by default stimulates the breakdown (resorption) of bones to salvage its components for other uses (use it or loose it principle). So yes, the skeletal system is directly connected to the brain (with credits to @anongoodnurse).