I suspect that was a misstatement - I haven't seen a proposal that avoids higher taxes without a large debt being written against something else (i.e. redirecting funding from the military). It is, however, accurate to say that personal out-of-pocket expenses wouldn't increase for most individuals. The cost per patient for government funded healthcare is expected to significantly undercut private costs per patient due to some efficiencies of scale and removal of some pretty big market friction points - so less money would end up being extracted from the economy to pay for an equivalent level of care.

In short - you can buy a banana for a dime or give the government a nickle and get a "free" banana.