...but if I add 'out' after the word 'held out' (David held out the box) does not make any sense, i.e I can not understand the meaning.
This is a literal application of the word "out." As in, David is holding the box outwards from his body -- he is stretching his arms away from his body while holding the box.
"Check it" makes sense to me. But I don't understand the meaning of "check it out".
This is an idiomatic expression. "Check it out" and "check it" can have roughly the same meaning, but "check it" usually implies a quick, routine check, rather than a thorough examination. "Check it out" usually implies that you are going to perform some level of thorough examination.
Can you check the mail?
I need you to check on Dylan and see if he's feeling well or not.
This is the normal application of "check it" and variants of that phrasal structure. The statements imply that the "checking" will take no longer than a moment.
Hey man, check it -- I got these new shoes.
This is a colloquial slang application of the phrase "check it" -- here the phrase is always used unmodified and is merely used to draw attention to whatever the speaker is referring to.
Hey, can you check out our server? The web service seems to be down.
Here, the phrase "check [it] out" is used. It is implied that a more thorough examination, lasting for an indeterminate amount of time, is required. Note that someone not familiar with networking or programming might have said "Hey, can you check our server?" not realizing that the issue might take much longer than a moment to narrow down and eventually fix.
It should be noted that many of these phrases have completely unrelated additional meanings.
For instance, the phrase "check out" is also the opposite of "check in," which means to register (in)/unregister (out) at a hotel, convention, or other location that keeps a strict record of its patrons.
"Held/Hold out" also means to stand one's ground, as in a conflict -- "Looks like we're going to have to hold out for another week before we get reinforcements."
Try not to get confused and always look at the context.