What would cause pain in the left side of neck when taking in a deep breath?

Accessory muscles: When you breathe in, you use your primary breathing muscle, the diaphragm. For a deep or strenuous breath, you also engage accessory muscles, including musles in the neck that "pull up" on your rib cage to help get more air in. If those muscles are sore, or if there is some other pain, inflammation, or irritation in the neck, then a deep breath can lead to pain in that area.

Myofacial pain: Often it is strain/sprain in the muscles of the neck. Might want to check if there is any abnormal palpable finding in the area of pain.