The Soviets were afraid that Able Archer was a cover for actual nuclear attack, and they raised their readiness in anticipation. However, if I've understood it correctly, they did it completely secretly and the West realized it only afterwards. So why the Soviets didn't go public or at least in someway communicate it to the NATO? Something like "We know what you are going to do, we are ready, its not going to work, back down", intending that if their "cover" is blown, NATO would not dare to attack. Or did they try something like this?
Also, during this time, the Soviets were also afraid of Pershing IIs being used in a decapitation strike. However, according to this very nice website, its range was only 1770 km, not enough to reach Moscow unless you drive it all the way to Denmark from the bases in southern Germany. Was this just the Soviets being unaware of the real range? What were actual targets for the missiles in NATO planning? And was the range of the missile a deliberate choice? It's Soviet counterpart SS-20 had over three times the range.
On Able Archer, the Soviets were not in the habit of going public about much of anything that was "real," so they certainly weren't going to reveal their strategic thinking to the Americans in any kind of official way. There was, however, a "backchannel" in which some Soviet sources did try to characterize the danger to at least on Reagan-era advisor (Jack Matlock). See document 18 here. Additionally, Oleg Gordievsky, who was a British asset in the USSR, similarly reported such fears. So there was some telegraphing of their concerns. (And there were some changes to Able Archer made, at least early on, to lessen these fears, such as not including the actual US political command in the exercise.)
On the Pershing II — the US and the USSR had a longstanding habit of asserting and contesting things like ranges, because they had both political and treaty implications. So the US frequently assumed the "worst case scenario" about Soviet capabilities (e.g., assuming they had longer ranges than they in fact did) and the Soviets did vice versa. In the case of the Pershing IIs, the Soviets said they did not trust the Americans' statement about the range. In such a situation there is little one can do to reassure the other party; it is relatively easy to prove that a missile does have a given range (you can test it at that range, or loft it, or whatever), but it is hard to assert it does not have that range (you can make technical arguments, but unless you are willing to let the other party look at every aspect of the missile — which the US wasn't — then you won't reassure them). Note that if they had deployed Pershing II's in West Berlin, they could hit Moscow at that range. But they deliberately did not do that.
I don't know what they would have aimed the Pershing IIs at — presumably strategic sites (e.g. deployed missiles, airbases, etc.) in Eastern Europe or Western Russia. The Soviets deployed huge numbers of IRBMs in Western Russia (their IRBM force was much larger than the ICBM force for most of the Cold War) and had some weapons deployed in non-Russian SSRs and satellite states as well (East Germany, Kaliningrad, Latvia, Ukraine, Belarus, Estonia, Lithuania, Czechoslovakia, Uzbekistan, Poland, and Kazakstan all had weapons in them at one point, and there were nuclear-capable bomber bases all over Eastern Europe). The same things that made a Pershing II a good decapitation weapon made it a good counterforce weapon as well (high-speed and accurate = you can put a lot of hard blast pressure wherever you want it to be, which is both what you need for an underground command bunker as well as an airfield, silo base, etc.).
I have been meaning to read Nate Jone's Able Archer 83 book — I haven't gotten to it yet (it's on the list...). I suspect it has more details on this than any other source. My main source so far has been the National Security Archive's Able Archer Sourcebook and Electronic Briefing Bulletins, and David Hoffman's The Dead Hand.