I don't get it. His prayer is permissable, but you are not allowed to hold it?!

Yes. His prayer is still valid (so he doesn't have to repeat it), but the action itself (holding the mushaf whilst following the Imaam) is not correct.

So should he do it or not?

Salamualikum.
^ No he shouldn't :S

He said:

So according to shaykh al-Albaanee (rahimullah), no.

But their prayer isn't invalidated

Yes.

The things that invalidate the prayer are well known, but the number differs according to differences among the fuqaha’. They are as follows:

1 – That which invalidates wudoo’, such as breaking wind or eating camel meat.

2 – Uncovering the ‘awrah deliberately. But if it is uncovered by accident and what is uncovered is only a little, or if a lot becomes uncovered but he covers it immediately, then the prayer is not invalidated.

3 – Turning away from the qiblah to a large extent.

4 – Presence of najaasah (impurity) on one's body or clothes, or in the place where one is praying. If he notices it or remembers it during the prayer and removes it immediately, then his prayer is valid. Similarly, if he does not find out about it until after the prayer is over, his prayer is still valid.

5 – Excessive continuous movement during the prayer for no essential reason.

6 –Omitting one of the pillars (essential parts) of the prayer, such as bowing and prostration.

7 – Deliberately doing an extra pillar, such as bowing.

8 – Deliberately doing some pillars before others.

9 – Deliberately saying the salaam before completing the prayer.

10 – Deliberately changing the meaning whilst reciting.

11- Deliberately omitting one of the obligatory parts of prayer, such as the first tashahhud. But if a person forgets, his prayer is valid but he must do the prostration of forgetfulness (sujood al-sahw).

12 – Deciding to stop praying.

13 – Laughing out loud. Merely smiling does not invalidate the prayer.

14 – Deliberately speaking. If one speaks by mistake or out of ignorance of the ruling, the prayer is not invalidated.

15 –Eating and drinking.

See: Daleel al-Taalib li Nayl al-Mataalib by Shaykh Mar’i ibn Yoosuf al-Hanbali (p. 34). And Duroos Muhimmah by Shaykh Ibn Baaz.

Source

So there are some things which are not correct and lessen the perfection of your prayer, such as smiling during the prayer, but do not invalidate the prayer. Allah knows best.

I find that i lose concentration when I dont have the mushaf wih me during taraweeh prayers.

Assalaamu Alaikum Wa Rahmatulaahi Wa Barakatuh

I find that holding the mushaf stops you doing a number of things such as looking at the place of Sujud, putting the right hand over the left hand, etc and there is that feeling of how and where to place the mushaf when you want to do sujud and or ruku'.

learn the quran by heart and you wont have a problem