Put the meaning into your own words.

The moderator asked speakers to discuss the plan without casting aspersions.

Later, a visitor began to insinuate that Celia had lost the venue because the staff disliked her work. His innuendo relied on raised eyebrows and a reference to what 'everyone had heard,' not on any identified source. Another remark cast an aspersion on her reliability by suggesting that she could never be trusted to keep a commitment.