Lighter bikes don't make you faster in steady speed riding on a perfectly flat surface. The hour record bikes are optimized for everything but weight.
But we are very bad at judging absolute speed without technical assistance, while being quite sensitive to acceleration. Therefore, a slight increase in acceleration for a given effort (or decrease in effort for a desired acceleration) is very noticeable, while an increase in absolute speed would not. So a lighter bike will feel noticeably faster (even though it isn't), while different aerodynamics (e.g. "faster" clothes that give a lot more improvement in absolute speed than a weight difference, even for weak riders) might give a clear advantage on the clock, but you won't be able to notice it while riding.
Light bikes just feel fast, very much so actually. It's a powerful illusion that only gets stronger because it is supported by those few specific situations (stop and go, races with tight corners, all that climbing road cyclists love) where the weight also matters objectively.