This sadly still happens even today and there are studies which have shown that those with clearly 'black' or minority names in general are less likely to get called back and less likely to get a job than someone who has the exact same CV with a white or ambiguous name.
It isn't exactly like this trend is reversed in non-white countries either, where being lighter skinned is often seen as an advantage and I've seen westerners hired to do jobs they probably are not all that qualified to do well (or, in my own field, I've seen white, British trained doctors offered a higher salary than brown or Arab British trained doctors for the exact same job and with the exact same credentials).
It is difficult to qualify what exactly 'white privilege' is, especially to those who don't feel its negative effects. The problem with privilege in general is that it is so multi-faceted, with the concept itself making people feel uncomfortable. People want to feel that everything they've done, they've achieved completely off their own back. Its uncomfortable to hear that there may be factors outside your control which have contributed and which may have contributed to others struggling.
The other issue it opens is that there are different levels of privilege. Simply by growing up in the UK, you are privileged. By speaking English as our first language, we are privileged. By working here, we are privileged. By having a 'Western passport', we are privileged. I'm a male, that comes with its own privileges. Its not clear cut though. Say you take the black son of wealthy Nigerian parents, who have enough money for him to go to Eton and then Oxford. He pays no fees and comes out of university debt free. By dint of the colour of his skin, is he less privileged than a white working class boy on free school meals and who's grown up in a single parent household? No, in his general life, he will probably go on to lead a far more 'successful' and wealthy life than the other.
I'm rambling now but white privilege does not mean every white person is racist. It does not mean every white person has an easy life and has stuff handed to them. It doesn't mean every single person of colour is going to be poor and unable to lead their lives. For me, it doesn't even mean that all white people should be trying to fight it. I get it, people live hard lives and have a lot of shit going on. It is most simply and fundamentally living life without having your colour be a real factor. Without having it affect your interactions with other people negatively. Without it being in the back of your mind as you interact with others.