The tournaments are not started automatically - that is the job of the tournament creator :-)

When you want to start it, click the button that says "Close for Signups" on the tournament page. That will take you to another page in which you will be allowed to start the tournament. Remember you have to click two buttons (one to close sign-ups and another to start the tournament) before it will start.

And also when the first round completes, if there is to be a 2nd round, that is your job to start as well. It will not start automatically ;-)