Indulging in the internet during work hours, sneaking past the staff's watchful eyes.
Literal
During-working [despite-なのに] staff [attributive-の] eyes [object-を] stealing internet [to-に] is-indulging-[continuous].
人の目を盗む ('to steal from someone's eyes,' meaning 'to do something while unseen, to sneak past watchful eyes') is a set idiom. 耽溺する ('to indulge, to be addicted, to wallow in') is a formal verb for excessive indulgence. 勤務中 ('during working hours'). のに marks the transgressive contrast.