。
She instructed us in singing technique.
Literal
She [topic-は] us [to-に] singing-method [object-を] taught [for our benefit-てくれた].
教授する ('to teach, instruct') is a Sino-Japanese verb with a noticeably formal, almost academic register — the noun 教授 also means 'professor.' Plain 教える would do the same job in everyday speech. 歌唱法 ('singing method/technique') is a compound of 歌唱 ('singing') + 法 ('method') — Japanese readily builds technical terms by stacking Sino-Japanese morphemes this way (思考法 'way of thinking,' 訓練法 'training method'). The benefactive ~てくれる frames the teaching as a favor done for the speaker's group, indicating gratitude.