This (very old) blog post got me investigating sources for the concept of "tzniut" ("modesty") and what is a blog for if not sharing this stuff. So this is the first source, from the prophets:
It hath been told thee, O man, what is good, and what the LORD doth require of thee: only to do justly, and to love mercy, and to walk humbly with thy God.
The phrase "walking humbly" in the refers to men and women (it may also be only men, but I’d have to know more about the context to be sure). It describes a general mindset or attitude towards life and the commandments. A person should be humble, not draw too much attention to himself/herself. This is reflected in clothes by not wearing anything too scandalous, too flashy or in other ways attention-grabbing. I’d say it extends to not wearing something that screams "expensive" or something that is making a radical political statement, etc. Men are just as susceptible to pride, arrogance and egotism as women, so the verse is relevant to both genders.
The possible extreme this verse can be taken to is obvious: absolute conformity. If it is not modest to stand out in any way at all, we’d all need to dress the same. We need to recognize that there is a difference between legitimate ways to express individuality and attention-seeking [though where to draw the line in practice may be very challenging, especially if teenagers are involved!]