Mysql Selecting from multiple tables
You probably meant AND Operator.OperatorID = RouteOperators.OperatorID

I'd suggest using the JOIN...ON (...) syntax, it makes the query more readable.

SELECT o.Phone,o.Email 
FROM Route r 
INNER JOIN RouteOperators ro 
    ON r.RouteNumber = ro.RouteNumber 
INNER JOIN Operator o 
    ON ro.OperatorID = o.OperatorID 
WHERE Route.Start = 7628 
OR Route.Destination = 7628


You might also want to consider introducing foreign key constraints.