When I wear my contacts, my eyes feel dry and get red.

Literal

Contact do-[when-すると] eye [subject-が] dried feeling [to-に] become, eye [subject-が] red become.

コンタクトする here verbalizes the loanword コンタクト ('contact lenses'). すると is the natural-consequence conditional 'whereupon / when one does X.' 感じになる ('becomes a feeling of') is a fixed expression for describing sensations. 目が乾いた ('eyes became dry') uses the past tense in the descriptive-state sense, modifying 感じ ('feeling') as a relative clause. 赤くなる ('become red') is the standard い-adjective → verb conversion via the く-form.