Isn't it something about a passive verb?
Yes it is! In formal grammatical terms, "I was stood" and "I was sat" are examples of the imperfect passive, the passive meaning that something happens to someone/something, rather than the active, where someone does something.
Obviously neither expressions mean literally (to use that word in its most precise sense - see posts above!) that someone or something picked you up and either stood you or sat you; the use of the passive in these examples is purely figurative / metaphorical. Thus "I was sat" carries more meaning than "I sat" - it could mean for example, that a waiter had shown you to a table, or that circumstances caused you to be sitting (quite often with the implication that the actual circumstances aren't relevant to the story). Sometimes there is an implication that the person telling the story isn't entirely happy with the circumstances, e.g., "I was stood in the immigration queue for hours!" = "They forced us to wait for hours in the immigration queue".
"I was stood at the corner when suddenly ..." could imply "It so happened that I was standing at the corner when ..." - with perhaps an idea that what happened next was beyond your immediate control.
So far as "The award was given to James by the organization" and "The organization gave the award to James", both of course are perfectly grammatical. The point the teacher was making might just have been a stylistic one - use fewer words when possible. Also there is a tendency to avoid the passive, if it is possible to use the active - it makes it livelier and more immediate, and often less pedantic. Compare, for example "Patrons are reminded by the Management ..." with "The Management remind Patrons ..."