As I lay back in the chair, the dental hygienist said to me, "Okay, please open your mouth."
Literal
Chair [on-に] lying me [to-に], dental-hygienist [gen-の] woman [subject-が] 'yes, mouth [obj-を] open-please' [quotative-と] said [explanation-のです].
A narration from a dentist visit. 椅子に横たわる is a relative clause modifying 私: 'me, who was lying on the chair.' 横たわる ('to lie down, to recline') is a formal verb. 歯科衛生士 ('dental hygienist') is a Sino-Japanese compound specific to the profession. お口 ('mouth') takes the polite お prefix — the kind of register a hygienist would use with a patient. 言ったのです is the explanatory/narrative のです past, framing the whole sentence as recounting what happened.