What do people mean when they assert that time doesn't exist?

They mean that they have trouble differentiating between time and the measurement of time.

The measurement of time is purely human invention.
Time itself, on the other hand, is a physical and necessary part of our reality.

I bet they show up for their appointments on time.

They mean that time is an illusion, lunchtime doubly so.  

It's not material thing that exists in the universe... It seems to just be the arrow of entropy.

My doctor put it this way “People assume that time is a strict progression from cause to effect, but actually from a non-linear, non-subjective viewpoint, it's more like a big ball of wibbly-wobbly, timey-wimey stuff.”

THAT IS SO FOOLISH..WE ALL HAVE CLOCKS..

https://www.youtube.com/watch?v=8Rf4zwPouIU

God said he is the Alpha and the Omega.  the beginning and the end Scientists don't believe in God even though time works in all of newton equations lol

Sounds to me like something an uneducated religious person would say.