There are some helpful mathsy explanations, but I thought perhaps this could use an intuitive example.
Suppose that you're investigating (perhaps for an insurance company) whether hair colour has an impact on crash risk. You look at the data, and at first pass you see that brunettes are 10% more likely to crash than blondes. But in the same data you see that brunettes are also more likely to get caught speeding. You do the controls to take out the effect of speeding, you'd find that the effect of hair on crash risk drops below signifance.
That would probably be an example of an inappropriate thing to control. It is likely that the fact that our brunettes speed more is the mechanism by which they're more likely to crash. As such, if you insist on zeroing out that mechanism, you're forcing yourself to see no effect even if it's obviously there. Intuitively, "Actually brunettes are very safe drivers considering how much they speed" is very obviously an unreasonable defence to make!
Conversely, suppose we look at the dataset again and find that people who are bald are 50% more likely to crash than those with red hair. But it also happens that the bald people in the dataset were typically older men, and younger women were underrepresented. Again you throw statistical controls at the situation and the effect disappears. This is probably a good thing to control, not least because your insurance company already asks about age and gender so you don't want to double count any effects. Again intuitively, saying "We already knew that age and gender are known to have an impact on road safety and on baldness prevalence. These data show bald young women are just as safe as hairy young women and bald old men are just as safe as hairy old men." seems like a very reasonable clarification.
(This example is entirely made up, and isn't an allegation that any particular hair type in the real world are dangerous drivers!)