Effectively, there are four explanations for this:
- First, the phrase is being used ironically. They say "With Due Respect", but they don't mean it.
- Second, it is being used as a half-hearted attempt to soften the blow, and hopefully avoid some of the retribution they would receive for just out-and-out insulting or contradicting someone in a situation where it would not be recommended to do so.
- Third, they mean it literally, and are implying that there is no respect due to whoever they are addressing.
- Fourth, they mean they normally do respect the person they're talking to, but disagree with one specific decision or statement that person made.