When an Oxbridge college awards you a place, they can link it to an exhibition or a scholarship. These awards confer on the winner a monetary allowance - very little these days, but quite a substantial sum originally. At the time, some decided to go to an Oxford college rather than a Cambridge one because the Oxford one awarded a scholarship. These awards also confer certain dining rights and residence rights. Scholars have preference when it comes to selecting rooms in college for example.
So basically, an exhibition/scholarship is an award (the scholarship being senior to the exhibition), based on academic merit, given to a limited number of students joining a college, or to those already at college on the strength of their results.
The "open" probably means it is not linked to a particular endowment, and comes out of general college funds.