It means "You must understand that I feel very guilty about this every day." Or in Chinese, 請你一定要瞭解，這件事沒有一天不讓我內疚（讓我良心「沉重」）。

you have to understand = (please understand) a day does not go by when = (everyday) that = (any thought, memory, past experience) does not weigh heavily = every day I think about that thing that ... on my conscience = (right and wrong) this speaker feels guilty about something.