Racism is more than bigotry. Racism is (1) the division of human beings into made-up categories and (2) systematic, institutional exclusion of humans beings based upon these made up categories.
The most insidious racism doesn't growl in your face, it smiles in your face while denying you equal rights and saying "we're just following the law" or even "we're just working here".
Racism does not require racist individuals to function, it functions independently of who's in the system. They're all "just doing their job". Many of them are nice people who look at you like you're unreasonable and don't understand why you're so upset.
Races don't really exist, they are manufactured. Racism is not the product of races - rather, races are the product of racism.
Many white people I know (most of them really good people) think that races are real but they don't matter. The truth is, races are not real, and it does matter.
The proper way to talk about racism shouldn't be "white people are excluding us", but rather "a system has been built that privileges white people whether or not they ask for it".
It doesn't make every individual white person automatically guilty, it just makes it more hard to see, just like it's hard for a fish to see the water it swims in. It takes listening to (and believing) the experiences of non-white folks, for them to start to see it.