When you walk into a room, you `enter’ it. You do not `enter into’ it. You normally `enter’ a room, building or country.
Examples: As usual, Chandru entered the room with a smile on his face. *The student wasn’t allowed to enter the country without a valid visa.
`Enter into’ is normally used in relation to `conversation’, `agreement’, `discussion’, etc. This is just one of the ways of using the expression.
Examples: After three days of fighting, the two companies entered into an agreement. *I hope to enter into a discussion with the leaders of the strike.