Williams helped Spain win the Euros over the summer and also chose to resist a move to Barcelona to stick with Athletic this season.
He told El Mundo: "Before the EC, people in Spain recognised me, but now it's a global thing. When we went to Italy to play against Roma in the Europa League, I thought people wouldn't know me there and suddenly I arrived and there were quite a few people waiting for me and saying my name or asking for pictures.
"It affected me. Everything is a little different compared to the life I lived before, but I accept it with great joy."
Williams has spent his entire career at Athletic Bilbao.
"In Bilbao, people tend to respect you a lot. In this city, the fact of playing for Athletic is a privilege and everyone treats you wonderfully, everything is praise and they don't overwhelm you. But in other places things get complicated.
"I always liked going to Madrid with my friends or my family and I could shop like anyone else, it went a little unnoticed. But the last few times I've stopped going to Gran Vía because everyone wants pictures, it's a big commotion and you can't even walk."
The 22-year-old was chased by several big clubs this summer, especially Barcelona - which was the closest to a deal.
However, Williams chose to stay at Athletic Bilbao. He is not considering a move at the moment.
"I don't think about it now. The fans are very happy that I'm staying here and I'm very happy to be here. Bilbao is my home, I want to continue to enjoy it and the truth is that I am very happy, he says and continues:
"I have always had things very clear and I think people see that. I think I'm doing really well and trying to take Athletic to the top in Spain and Europe, with that final in San Mames exciting us so much. That's the only thing that matters now."