Basically an intention is made up in your mind. As for example if you want to pray a prayer you prepare yourself by thinking that you might need to perform wudu' etc. so the process basically starts in the mind, with the heart (which might guide you to do good things). In Arabic we say the niyyah is in our heart -also meaning inside ourselves-!
Some scholars or madhabs say it is ok saying your niyyah to remind yourself, but saying (reading) it can become an issue, as once you start saying you may make things more complex than they are:
For example you want to pray 'Isha':
In your mind after performing your ablution (wudu') you might go to a mosque (with the intention to pray), once at the mosque you might join the congregation (with the intention to pray in a congregation and gain the rewards). Now it can become excessively complex if you follow some guide lines of some madhabs, for example they say you must fully express you niyya, that means you might say to yourself, I'm intending to pray the 'isha' (fard) in 4 raka'a with this (following the) Imam in congregation, so here many mistakes can appear, for example instead you say you'll pray 'asr or forget one of those expression parts, but is that really a correct niyyah or did the Propeht () do so?
No a niyyah by definition is the intention to do an act at the very beginning of it, so it is basically a simple thought of the kind "Hey I'll do that now for the sake of Allah" followed by "Allahu akbar" (the takbir to start in the example of a prayer).
The same applies for other worships like fast, it might cause trouble to say in the night before fasting: "I intend to fast tomorrow", as you may create doubt, as the sunset declares the beginning of the day -in the view of Islam-. So you may ask yourself is my niyyah correct, as tomorrow in the view of Islam would start after the next maghrib (sunset), while I want to start fasting from the actual fajr on meaning by tomorrow the day which will start at 0:00 ... So the best is to have in mind "I'll fast this day incha' Allah" at least a few minutes before the adhan of fajr and everything is fine ...
And if the Propeht () would have done it this way he would have thought it to the sahaba and we should have narrations supporting that.
There's no backup for this excessive or exact formulation nor for saying the niyyah, but it can be helpful as a remainder no more nor less. The niyyah itself is done by a simple thought, so why make it so complex?
قال شيخ الإسلام كما في الاختيارات: ومن خطر بقلبه أنه صائم غداً فقد نوى
Ibn Taymiyyah said: If you had in mind that you are fasting tommorrow then you already have made your niyyah.
Therefore some other scholar with a rather orthodox view, say saying the niyyah or speaking it out is bid'ah. But as I've said before as long as it helps you to concentrate on what you want to do there's no harm in doing it, once it is too complex as shown in my example above it is meaningless, too long and can create doubts, as if you said 'asr instead of dhor you might stop your prayer or redo your niyyah, but in fact in your mind you already have chosen the correct prayer and made the correct niyyah, but did a simple mistake while speaking it out.
Allah said in (98:5).
And they were not commanded except to worship Allah , [being] sincere to Him in religion, inclining to truth, and to establish prayer and to give zakah. And that is the correct religion.
So the most important thing is doing things sincerely not how one omits the niyyah or expresses it.
In ahadith niyyah is also quoted for example in sahih al-Bukhari and sahih Muslim on the authority of 'Omar ibn al-Khattab(), in sahih Muslim on the authority of abu Hurairah (), in sahih a al-Bukhari on the authority of ibn 'Abbas () and others.
so where is speaking out the niyyah quoted here?
And always have in mind the hadith saying:
Verily Allah does not look to your faces and your wealth but He looks to your heart and to your deeds. (sahih Muslim)