"They got dropped on me out of the blue, and I barely know anything about them." 'Dropped on' is a phrasal verb like 'drop in" or "drop by" - meaning to visit someone without calling first - you just appear, like a surprise visit. But drop in and drop by are inseparable - we always 'drop in on someone (James)' or 'drop by someplace' (James' house) 'Drop on' would have a similar meaning but it is separable - My boss 'dropped (a ton of work) on' me at the last minute. Or as in your sentence where it is not separated 'They (whatever they are) got dropped on me'. So it still has this sense of being unexpected but these phrasal verbs are used differently. The other expression in your sentence is 'out of the blue' which also means that it was unexpected. We also say 'out of nowhere' to mean unexpected, sudden or by surprise.

That sounds like slang for surprising someone suddenly with something. Dropped on me could also be "sprung on me" but both of these phrases are extremely informal and some places may not use these. Young people would probably be the only people to use that phrase.

It probably means that they were suddenly given to me, and I hardly knew anything about them. 'Dropped on' can be literal, but I suspect in this case it isn't. I guess that it is referring to tasks, or jobs, or something similar.