Why did Montezuma allow Cortes into the capital?

Because he thought he was the god Quetzalcoatl coming back to reclaim his throne.

According to the Aztecs, Quetzalcoatl was king in the city Tollan (Tula) until his brother Tezcatlipoca, through treachery and black magic managed to make him step down and go into exile. The Aztecs believed that Cortes was Quetzalcoatl coming back from that exile.

PD: Some people claim that this was just a legend created by the Aztecs to find an excuse for their quick fall. This people say that Montezuma II allowed Cortes into Tenochtitlan just because he was unsure of how to act against this newcomers.