This is a pretty standard clause on many new housing developments and it is for the reason in Loonytunes first sentence.
It means that you need both the Developers permission and the normal LA Development Control permission.
In the first instance, phone the Developer to find out how much it wants for processing this - I remember paying �25 in about 1991 to do this. As you quoted yourself - such approval will not unreasonably be withheld.
Having established how much, THEN get PP before reapplying back the Developer for permission. The reason for doing it this way is because if LA Development Control want you to modify the plans, you would technically have to go back the Developer with the modified plans again - plus another fee.
Don't be tempted to skip this convenant - my neighbour did and when he wanted to sell the buyer's solicitor picked it up. He had to pay �250 for retrospective release from the terms of the convenant.