。
She takes pride in her cooking skills.
Literal
She [topic-は] cooking [possessive-の] skill [object-を] is-proud-of.
料理の腕 (りょうりのうで, 'cooking skill') uses 腕 ('arm') figuratively to mean 'skill, ability' — similar to English 'arm' being used for capability in some contexts. 自負する (じふする, 'to take pride in, to be confident about') is a somewhat formal suru-verb.