Typically (but again there might be exception depending on who is your datacenter supplier), the tenants buy optical fiber running between their rack or cage to the Meet Me Room.
All the tenants do this, so they actually meet in the MMR. When they want to interconnect their infrastructure together they ask the datacenter staff to install a cross connect between their respective position they bought in the MMR.
Basically it's a row of racks full of patch panels.
A picture is worth a thousand words so have a look below.
The yellow optical fibers are coming from all the tenant's infrastructure and ending in one of the rack pictured. So when the datacenter staff know both positions to be interconnected together, they run another piece of optical fiber between these two positions.
Since a Meet Me Room is usually a sensitive part of a datacenter infrastructure, access to this room is naturally restricted to datacenter staff.
Some service providers are so big and they handle so many cross connect that they might have their own MMR in a given datacenter in order to reduce the costs associated to it.