A hypothesis is that:
The common ancestor of all Sino-Tibetan languages is associated with the Cishan Culture 磁山文化 (c. 6500 – c. 5000 BC) on the eastern foothills of the Taihang Mountains (Sagart et al., 2019).
The proto-Sino-Tibetan language began to diverge due to the expansion of the Early and Middle Yangshao Culture 仰韶文化 (c. 5000 – c. 3000 BC) along the middle reaches of the Yellow River, and eventually split into at least two branches around 6,000 (Zhang et al., 2019) or 8,000 (Sagart et al., 2019; Zhang et al., 2020) years ago. The Yangshao Culture is considered a descendant of the Cishan Culture (and a few other cultures, e.g., Peiligang, Laoguantan).
The Tibeto-Burman branch (including modern-day Tibetan, Qiang, Lolo-Burmese, Karen, etc.) is associated with the Majiayao Culture 马家窑文化 (c. 3300 – c. 2000 BC) in the upper Yellow River region (Sagart et al., 2019; Zhang et al., 2019). The Majiayao Culture is considered as the result of a westward migration of the Yangshao Culture population.
The Sinitic branch (including modern-day Chinese, Bai, Caijia, etc.) is disputably associated with the post-Yangshao culture(s) in the middle Yellow River valley areas, sometimes lumped under the umbrella term "Longshan Culture 龙山文化" (the term in a narrow sense refers exclusively to the Shandong Longshan Culture). Archaeological sites involved are Miaodigou II 庙底沟二期 (c. 3000 – c. 2600 BC), Wangwan III 王湾三期 (c. 2400 – c. 1900 BC), Keshengzhuang 客省庄 (c. 2600 – c. 2000 BC), Taosi 陶寺(c. 2300 BC – c. 1900 BC), Xinzhai 新砦期 (c. 1900 – c. 1700 BC), Xiaqiyuan 下七垣 (c. 1800 – c. 1600 BC), and so on. These cultures as well as the Sinitic languages their people spoke embody distinctive non-Sino-Tibetan features from the proto-Austro-Tai (?) speaking (Sagart, 2008) Dawenkou Culture 大汶口文化 (Wen et al., 2016). The Shang Dynasty (c. 1600 – c. 1045 BC) would then assimilate different Sinitic dialects into the highly homogenous Old Chinese (in its earliest form).
To sum up, the pre-Shang Sino-Tibetan languages went through the following paths of evolution:
Cishan 磁山 (proto-ST) --> Yangshao 仰韶 (proto-S+proto-TB) --> Majiayao 马家窑 (proto-TB)
Cishan 磁山 (proto-ST) --> Yangshao 仰韶 (proto-S+proto-TB) --> "Longshan 龙山" (proto-S)
Beixin 北辛 (proto-AT?) --> Dawenkou 大汶口 (proto-AT?) --> "Longshan 龙山" (proto-S)
Note that Austro-Tai as a language family is still under debate.
Here is a map (Sagart et al., 2019) depicting the hypothesized homeland and dispersal of the Sino-Tibetan languages.
Sagart, L., Jacques, G., Lai, Y., Ryder, R. J., Thouzeau, V., Greenhill, S. J., & List, J. M. (2019). Dated language phylogenies shed light on the ancestry of Sino-Tibetan. Proceedings of the National Academy of Sciences, 116(21), 10317-10322.
Sagart, L. (2008). The expansion of Setaria farmers in East Asia: a linguistic and archaeological model. In Past human migrations in East Asia (pp. 165-190). Routledge.
Wen, S. Q., Tong, X. Z., & Li, H. (2016). Y-chromosome-based genetic pattern in East Asia affected by Neolithic transition. Quaternary International, 426, 50-55.
Zhang, M., Yan, S., Pan, W., & Jin, L. (2019). Phylogenetic evidence for Sino-Tibetan origin in northern China in the Late Neolithic. nature, 569(7754), 112-115.
Zhang, H., Ji, T., Pagel, M., & Mace, R. (2020). Dated phylogeny suggests early Neolithic origin of Sino-Tibetan languages. Scientific reports, 10(1), 1-8.
According to Sagart et al. (2019), Matisoff's Tibetan origin hypothesis mentioned in the comments is untenable because it "lack an archaeologically and demographically supported account of the family’s expansion".