There are no shoes on the dirty floor, meaning Ayanami enters the room with her shoes on.
There is a mountain of mail even on the shoe rack. Shinji’s feet can be seen entering the apartment.
SHINJI:“Excuse me! This is Ikari… Ayanami! I’m coming in!”
Shinji is about to step up into the room with his shoes on when he stops.
thewayneiac: As you might have heard, the Japanese never go past the entranceway without removing their shoes. Shinji is more conciencious of Rei's home than she is herself.
Mr. Tines: The state of cleanliness of the floor doesn't look good. I think Shinji's socks will get the worse of the encounter.
UrsusArctos: Taking off your footwear when you enter is common across nearly all East Asian cultures. On the practical side, walking into a house with footwear brings in dust and dirt from the outside. And there's the issue of footwear being made of animal products (ritually impure).
Dr. Nick: It's certainly not just a Japanese thing. I wonder if there's a world map floating around showing in which cultures shoes are commonly worn indoors and where they're usually removed.