The profile information states that the profile contains a 'device enrollment challenge'. A device enrollment challenge requires connection to a device enrollment server—not Apple's servers, but another server that the payload is informed of.
When you press Install and enter your passcode if appropriate, all the payloads that are contained within a mobile configuration profile are activated. The device enrollment challenge payload attempts to connect to the aforementioned server at this point. It is at this stage that the error shows: "The request timed out". This is to inform you that the server that the payload is attempting to contact is unavailable and the request has timed out.
To fix this, you'll need to allow whatever is preventing the connection to succeed. The problem may not be the device though—the server itself may be unavailable. Unfortunately, a 'request timed out' error is very broad and could be caused by many different reasons, but the basis is that the server (again, not Apple's) is unavailable to the device requesting the connection.