What determines when twining plants twine?
Thigmotropism

This is a special case of tropism, called thigmotropism. In botany, tropism is a modified growth in response of an environmental stimulus. The most famous tropisms are:


Phototropism: plant growth in response to a light stimulus
Gravitropism: plant growth in response to gravity


In the case of thigmotropism, the growth is modified in response to touch (from the greek θιγμός). Curiously, this phenomenon was studied by Darwin in his late life: Darwin, C. (1865), On the Movements and Habits of Climbing Plants.. Journal of the Linnean Society of London, Botany, 9: 1–118. doi:10.1111/j.1095-8339.1865.tb00011.x

The mechanism of thigmotropism can be compared to that of phototropism: in phototropism, the shaded side of the stem apex has a higher concentration of auxins compared to the sunny side and, therefore, grows more quickly, and the result is that the plant grows towards the light source (positive phototropism). 

In thigmotropism, there are mechanic receptors in the epidermis which detect touch (mechanoreceptors), possibly membrane proteins. Then, after detecting touch, there is a signal transduction somehow, which "informs" the cells opposite to the touching side about the stimulus, and those cells exhibit an increased rate of growth. 

The mechanism may be similar to that of phototropism, also depending on auxins. According to Scorza et al. (2011):


  Ultrastructural studies revealed that most thigmotropic and thigmonastic movements are caused by differentially changing cell turgor within a given tissue. Auxin has emerged as a key molecule that modulates proton extrusion and thus causing changes in cell turgor by enhancing the activity of H+ATPase in cell membranes. (emphasis mine)


The actual details regarding the mechanism are not well understood, and they are way less understood than those of phototropism or gravitropism, for instance. The result, however, is that the touching side grows less that the opposite side:



If you combine thigmotropism and negative gravitropism (the growth against gravity), the result is that the climber grows up going around the support, in an upwards spiral:



A detailed explanation can be found here: Plant tropisms. 1st ed..

Sources:


Scorza, Livia Camilla Trevisan, and Marcelo Carnier Dornelas. “Plants on the Move:  Towards Common Mechanisms Governing Mechanically-Induced Plant Movements.” Plant Signaling &amp; Behavior 6.12 (2011): 1979–1986. PMC. Web. 14 May 2017.
Gilroy, S. and Masson, P. (2008). Plant tropisms. 1st ed. Ames, Iowa: Blackwell Pub