He asks us to question not whether something is an X or a Y, but rather to get "meta" and start examining what makes it possible for us to go through life assigning things too ontological categories (X or Y) in the first place. Because the mechanism of ontology -- that everything has a "being", that it "is" something -- is not a given
This is exactly where anyone coming to Heidegger for the first time ought to start. Unfortunately, for many people it's also infuriatingly difficult to unpack, due to Heidegger's style. For example:
"The discussion of Being as just that -- Being -- still speaks an inadequate language, insofar as, in our perpetual references to Being itself, it is addressed with a name that continues to talk past Being as such. In making this remark, we are voicing the assumption that Being -- thought as such -- can no longer be called 'Being.' Being as such is other than itself, so decisively other that it even 'is' not."
This obscure ditty[1] is from Nietzsche, Vol. IV. The key, as you point out, is to focus on the 'is' and, equally importantly, the 'as'. Replace "Being" with "the hammer" or some other object and things start to clear up, at least a little bit. The insight is that that thing in the carpenter's hand exists 'as' a hammer (for the carpenter, for you, for me), but it 'is' not a hammer; the 'is' of a thing is simply an 'as', a way of being in a world. A world or a way of being in it can change, and then the hammer 'is' something else.
A seemingly small insight, to be sure, but it's Heidegger's point of departure, and has a wide range of practical applications in later thinkers, even ones working on the sciences (e.g. Bruno Latour and others doing "sociology of science"). Deconstruction -- destruktion is Heidegger's word -- is on balance the practice of resisting what is given, of not falling "into the grip of ideas already decided beforehand"[2]. The clearest statement of this idea in Being and Time is I think in §32, which starts on H149. This is the starting line for basically all contemporary theories of interpretation, from Gadamer to Ronald Dworkin.
1. Of course there are reasons for Heidegger's writing the way he does. Whether or not they're good ones is a whole 'nother discussion.
2. "Language", from Poetry, Language, Thought.
What are the reasons? They'd want to be pretty good ones :-)
Well, it's a deliberate change that he makes later on in response to a feeling that people (not only critics, but admirers like Sartre) haven't understood his work. In "Letter on Humanism" he says of the relatively straightforward language of Being and Time:
I have learned to see that these very terms were bound to lead immediately and inevitably into error. For the terms and the conceptual language corresponding to them were not rethought by readers from the matter particularly to be thought; rather, the matter was conceived according to the established terminology in its customary meaning.
In short, he tried to say something new using all the old terms[1], and that allowed readers to assume something about the meaning of those terms rather than work it out for themselves. So, to get his point across, Heidegger starts looking for a new way of saying, one without shortcuts, one that forces readers to think things through.
Did it work? In a sense. Of course it also ensured that a lot of people will never bother reading Heidegger. For me, I simply treated Heidegger as a foreign language and learned to speak it. Once you're over the hump, he's not wildly difficult, and the reward, I think, is not unlike what you get out of learning, say, a functional programming language. It gives you new eyes, rather than just showing you some new thing.
1. 'Term' is often used with a special meaning in Heidegger. From On the Way to Language (with a few bits omitted): "Everybody sees and hears words. They are; they can be like things, palpable to the senses. To offer a crude example, we only need to open a dictionary. It is full of printed things. Indeed, all kinds of things. Plenty of terms, and not a single word. Because a dictionary can neither grasp nor keep the word by which the terms become words and speak as words."