To "open the floor" is a set expression: when someone is speaking in public (i.e. before an audience), when they are done with their speech, they will 'open the floor to questions', meaning that they are taking questions from the audience. Therefore the "floor" (the area in which the spectators are seated) becomes "open" (so that they can ask questions if they like).

Hello Elenvital: Large meetings for organizations are conducted (as with entertainment) from a stage. Below the stage is an extended "floor" where the majority of the people gathered stand. From the stage, the key persons or the most important persons sit or stand to present certain ideas with clarity. Usually, after some public discussion and explanation, the people on the stage will accept the questions from the rest of the gathered attendees who stand on the "floor". (They may sit in an auditorium, but that is still referred to as the "floor". If you are on a stage, and I am a person with a question, I will wait politely until the host of the proceeding opens the meeting to Questions----From---The---Floor. Then I will raise my hand and wait to be called upon. Sometimes, people line up to wait there turn to speak into a microphone so that everyone can hear their question.