What is it called when someone always starts their comments with "I know it seems silly, but..." or "This might sound stupid, but..."? What is that type of prefacing called? Is there some specific term?

Self-effacing would be one term.

Insecurity, apologetic, and ^^. The assumption by the speaker is that “It is silly” or “This does sound stupid.”

It’s a disclaimer.

On the internet, it is very difficult to properly communicate, and many people use polite disclaimers (thank you jeruba)to help difuse possible misunderstandings.

@Jeruba – I just saw this question – that is exactly what I was thinking.

A disclaimer, you also see them (or not) at the end of infomercials or at the end of an application to join a new web community or utility…

Legal disclaimers are not the only kind of disclaimer.

obviously.

Could it also be self-depreciating?

These prefaces are self-deprecating when a person uses them in spoken speech – which the original poster is referring to.

When a person continually begins their spoken sentences with self-deprecating disclaimers, people often get annoyed.

Here is a source that shows you how to coach such people:

The self-deprecator

We’ve all been the victim of self-deprecation before. The times when we have lost our confidence in ourselves and used language that conveyed to others we were incapable of doing things. Now this is okay as we quickly recognised it or our friends, family and colleagues would ask us why we were putting ourselves down and hopefully we would get out of self-deprecation mode and regain our confidence.

For some people they have developed the art of self-deprecation and function very well. They draw people in and make people feel sorry for them, thereby offering help when someone is putting themselves down, this quickly becomes tiresome and the person who is helping all the time will start to avoid the person putting themselves down.

Self-deprecation can also be an indicator of some form of mental illness; severe depression, and stress. It can also be an indicator of some kind of trauma in someone’s life such as abuse, in the past and in the present. Many women who suffer from domestic abuse are prone to self-deprecation. Therefore, we as colleagues, friends and family have a duty to try to understand what is going on in someone’s life and help them through difficult times.

Dealing with the self deprecator

There are ways to deal with people who put themselves down all the time. Here are just a few things you can do:

Point out to the person when they are using self deprecating language
Ask if there is anything they want to talk about
Point out their strengths
Stop them from apologising all the time
Never put them down in front of people

The cycle of self-doubt

If you constantly say to yourself you can’t do something, guess what, you won’t be able to do it. If you don’t do things you will sink further into the ‘can’t do’ attitude and try even less activities. The less you try to do the lower your self esteem will become. This is a vicious circle and is hard to break if not recognised.

Stopping the cycle and gaining confidence

If you are someone who puts yourself down all the time there are a few things you can do to try and stop the cycle and regain your self confidence

Focus on your strengths and do more things around them
Work on smaller goals which are achievable
Do not compare yourself to someone else
Think about your successes in life (everyone has success stories)
Learn from someone you admire (Don’t compare, just learn i.e. how do they react in certain situations, how do they walk, how do they talk)
Learn to talk positively to yourself
Recognise the demons of self consciousness and stop them in their tracks by talking positively and remembering your successes
Congratulate yourself often, even with small jobs
Build your confidence slowly and learn from each success

@squirbel GA I have to read the question better, I missed the part about the spoken word

qualifiers also works. disclaimer is probably the better term.