There seem to be two parts to this question.
First: "Have US drone strikes killed thousands of non-combatiant civilians between 2001 and 2013?".
The answer is probably, though likely not the full 6,000 you mention. The best source for this is The Bureau of Investigative Journalism's database. The Bureau uses a well designed and publicly avaliable methodology to count the number of people killed by drone strikes year to year. In total, they count approximately 3,500 people killed by US drone strikes. Of these, only between 500 and 1000 deaths could be confirmed as "civilians", though the organization makes note of the fact that the US government may overuse the term "Militant" when describing causalities, and many victims cannot be confirmed either way. A further ~1,250 civilians were confirmed injured. A visualization of this database is available here.
It is possible (perhaps even likely) that the actual number of deaths is higher. This data establishes only a lower bound. A US senator, presumably with inside knowledge of the program, claimed in 2013 that the actual number (of both military targets and civilians) killed exceeds 4,700.
Second: "Is the US killing these civilians to terrorize the associated populations?"
The answer is much more subjective, but I think probably not.
First, I define terrorism according to what I believe is the conventional definition. A terror attack is one that explicitly targets civilian, rather than military, targets. The intention behind such an attack is to inspire terror in the civilian population. I do not think the attacks conform to this definition, but it is a contentious issue, in part because of the subjective nature of the terms involved.
The Guardian has a nice summary, including graphics, up here, based on the report of Human Rights advocacy group Reprieve, which is in turn based on the Bureau's database pointed at above. Approximately 1,000 reported civilian deaths by drone strike are attributed to attempts by the US government to kill specific individuals, associated in one way or another with the US government's "War on Terror". For instance, attempts to kill eminent figures in Al Qaeda or the Afghanistani Taliban. The human rights organization points out that an average of 28 civilians are killed for each military target, which seems like a large number. However, it seems indisputable that the targets of the strike were military in nature.
On a more editorial note: it is probably not in US interest to inspire terror in the civilian populations of Yemen or Pakistan. Both groups are already hot spots for recruitment into organizations the US is trying to eliminate. Presumably then, these deaths are examples of collateral damage. Of course, there is a moral question of whether the intentions of the US government negate the effects of their drone strikes, but I think it's hard to ascribe terror as a motive here.