This review by Qian-Qian Li et al 2013 found there’s emerging evidence to suggest the way Acupuncture works is to alleviate the ‘fight or flight’ response and restore balance within the brain and in the body.
“Emerging evidence indicates that Acupuncture treatment not only activates distinct brain regions in different kinds of diseases caused by imbalance between the sympathetic and parasympathetic activities but also modulates adaptive neurotransmitter in related brain regions to alleviate autonomic response.” 

A 2018 Systematic Review of 13 research papers by Amorim et al, showed Acupuncture may be effective for anxiety disorders.
“Overall, there is good scientific evidence encouraging Acupuncture therapy to treat anxiety disorders as it yields effective outcomes, with fewer side effects than conventional treatment. More research in this area is however needed.”

While more studies are needed to confirm these results, this Systematic Review of 30 studies by Shergis et al in 2016 found Acupuncture may be effective for insomnia.
“Acupuncture compared to sham/placebo and pharmacotherapy showed statistically significant results”