Ne pas y aller avec le dos de la cuillère, literally “not going at it with the back of the spoon”, means to act bluntly and deliberately, without moderation. It is often used when someone is very honest about an issue. The closest expression in English would be “not mincing one’s words”.
This expression appeared in the 19th century. Its origin is not very clear but it it linked to the fact that it is inefficient to eat soup with the spoon turned upside down, that is to say with the back of the spoon.