Your taxes may go up, but the claim is that the amount your taxes go up will cost less than the healthcare premiums/out-of-pockets/copays/prescriptions/etc. you

Your taxes may go up, but the claim is that the amount your taxes go up will cost less than the healthcare premiums/out-of-pockets/copays/prescriptions/etc. you have to pay, thus the average individual's net cost for healthcare is lower.