Saturated oils are oils where every carbon is sp3 hybridized and attached to two other carbons and two hydrogen. An unsaturated oil features a pi bond and a sigma bond between one or more carbons. This sigma bond interacts with Br2 by way of an addition reaction, the double bond is broken and two bromine are added to the carbon chain. The resulting structure is colorless so in a way the oil absorbs the colored Br2 into a colorless molecule. So, the more Saturated an oil is, the more Br2 it can accept and that's why Br2 can be used to detect the presence of satiated oils. While adding Br2, the solution will stay colorless as long as there are double bonds to accept it.
Hope this makes sense, if you've talked about reaction mechanisms this should be pretty straightforward.
