Persistent spell is a prerequisite for DMM: Persist, and extend spell is a prerequisite for that.
... Not clear wording here.
Let me be super duper over detailed.
Extend Spell is required for Persistent Spell. Persistent Spell is Extend spell on steroids, so this makes sense.
Persistent Spell is required for Divine Meta Magic: Persistent Spell. Divine Meta Magic lets you use a Meta Magic feat better than normal. You can't use a meta magic feat better than normal unless you have that meta magic feat initially. So this makes sense.
Divine Meta Magic lets you use Meta Magic feats for free... You don't have to trade higher level spells for lower level ones which have been buffed. Using Meta Magic feats without some way to make them free is generally considered very stupid, because higher level spells are generally so much better than lower level spells that Meta Magic feats cannot help the lower level spells compete.
The catch: Divine Meta Magic costs Turning Undead attempts.
Why you don't care: You're trading nothing in return for a lot. Turn Undead sucks major ass. Undead enemies are very rare unless you're playing in an campaign with an almost exclusive focus on them. Free meta magic is more powerful. In the case of Persistent Spell, it is much more powerful.
Clerics get (3 + their Charisma modifier) Turn Undead attempts per day, if I recall correctly. Most Clerics don't have a monster charisma, so 16 Charisma or 14 or less. We'll say they get a +3 Charisma modifier. That's only 6 Turn Undead attempts per day. Let's say you want to use Empower Spell with Divine Meta Magic. Empower Spell usually bumps a spell up two spell levels. Divine Meta Magic costs however many spell levels you would normally bump up a spell +1. So using Divine Meta Magic: Empower Spell would cost 3 Turn Undead Attempts, meaning you can use it exactly twice per day.
That's not very helpful. We want to use a Meta Magic feat which is very cost efficient. Persistent Spell in this case is very efficient. It gets around wasting a lot of Turn Undead attempts on just one spell by making that spell last all day.
But Persistent Spell costs a lot of Turn Undead attempts. If you follow the Errata (I wouldn't suggest following obscure Errata at all; just go off what you have in your books), it costs a +6 spell level adjustment. If you just read it out of Complete Arcane, it's just +4 spell level adjustment. Either way, normal clerics would only be allowed one or zero use of Divine Meta Magic: Persistent spell each day.
This is where Extra Turning comes in. +4 Turn Undead attempts per day helps out a lot. More Persistent Spell'd spells is good. Worth a feat even.
And then there are methods of gaining more Turn XYZ attempts per day to fuel Divine feats. Turn Vermin from Drow of the Underdark works for Divine feats. Then getting Turn Dragons from Dragon Magic. Then getting Turn Undead from a Cleric prestige class (Like Inquisitor from Complete Divine). You can only use the Turn XYZ pools which aren't Turn Undead that specify they work for fueling Divine Feats. Otherwise they don't because Divine Feats usually only work for Turn Undead.
So getting the Fire, Earth, Water, or Air domains from the PHB won't give you more Divine feat uses per day, sadly.
It should be noted Extra Turning grants +4 Turn XYZ attempts to ALL Turn XYZ pools. If you have Turn Vermin and Turn Undead, you get +8 Turning attempts. If you have 3 pools of Turn XYZ, you get +12 Turning attempts. If you have a lot of Turn XYZ pools, taking Extra Turning is a very good idea.