She boasts that she's good at cooking.

Literal

She [topic-は] cooking [subject-が] skillful [quotative-と] is-boasting.

自慢している (じまんしている) means 'is boasting/bragging.' The quotative と marks the content of the boast: 料理がうまいと. 自慢する is a useful suru-verb for describing someone proud of their abilities.