What's the difference between a verb like read and read out or shout and shout out and so on? How does "out" change the meaning of verbs?
Usually when applied to a verb involving speech, out involves the addressing of a group, normally a non-specific group, as in "anyone who will listen."
John spoke out when he saw injustice being done.
Mary shouted out for help.
The bailiff read out the charges against the defendant.
In other words, all who could hear were being addressed in each case.
Don't confuse "read out" with readout, however. As a noun, a readout is simply "a visual record or display of the output from a computer or scientific instrument." [NOAD]
It's part of a Phrasal Verb, so it can have several different senses. Which sense is intended -- or understood -- depends on the verb involved, and the idiomatic context of its use. Almost all Phrasal Verbs are idioms, after all.
One sense that is available with out is, simply, 'outside, outwards', as in move out, sing out, cook out. Another is 'to completion', as in burn out, muck out, work out. More details, and more particles, and more phrasal verbs, are explained in the two Phrasal Verb links above.
The combination of read and out is called a phrasal verb. For read out, you can deduce what this phrasal verb means by knowing what read and out mean, as the other answers say. However, not all phrasal verbs are so easy.
For example, speak and talk are nearly synonyms, but to speak out means "to talk freely and fearlessly," while to talk something out means to "discuss something exhaustively" or to "resolve or settle something by discussion." (All definitions taken from the Free Dictionary.) And this isn't the same as talking somebody out of something.
To call something out or call out something is to say it very loudly, but to "call out the guard" ("the troops", "the Marines", etc.) means to "summon them", and to call somebody out is to "challenge them" (originally to a duel, but now used more generally).
You basically have to learn the definition of all these phrasal verbs, at least the ones where the meaning isn't clear from the meanings of the two components. You should think of the meaning of the original verb and the preposition/adverb as more of a guide or a mnemonic than as a way of deducing the meaning.
At times calling the use of out with a verb a phrasal verb is justifying poor English. I am not questioning the legitimate phrasal verbs given as examples above: call out, burn out, muck out, work out. We test a mattress, not test out a mattress. You can switch your style. You can switch it up. Switch up your style? Switch does not need a modifier in that case. Test and so many other words, do not need a modifier. A modifier is to alter or add something.
Superfluous words do not make us seem smarter. There has been an increase in the use of up, and out, since everyone, and anyone, began publishing online without any peer review.
I'm not sure about your example of testing and testing out. They mean different things to me. I don't think we do tend to test a mattress - that's usually the manufacturers job? Or perhaps it's something I might do when I'm in a mattress shop (personally I'd say 'try'). I test a mattress out at home (ie go for a sleep, or whatever else people do on mattresses). It's the same with shoes, I try them on in a shop and then test them out on a walk/run/game.
As for 'up' - the use of 'up' adds meaning, adds precision, often showing a completion. There's a difference telling my son to 'eat' and to 'eat up.' To switch up styles is a complete change, a transformation.