'Trade' is a tricky word in this period - what we're talking about is the realignment of long-distance networks of exchange. Cline is right to flag this up and in fact makes much less of it than most scholars - in the general narrative I've sketched, the primary process behind what happens in the final LBA is that technologies and institutions change so that goods are no longer going where they used to go by the means that they used to. An awful lot of the rest follows from that.

The definingly 'Bronze-Age' way of exchange is large-scale 'gift' exchanges between rulers. 'Trade' isn't quite the right word - it's primarily about status, because giving grandiose gifts proves your wealth, generosity and power (the subtext is 'look how great I am, that I can send you these awe-inspiring treasures as if it's nothing to me'), and you're expecting a return though not calculating exactly 'how much' you've received and trying to make a 'profit' (in fact, you're trying to make a 'loss', because being unable to equal or exceed a gift you've received is a major humiliation for your friend/rival in the other state). We have a little evidence for how this happened - most notably in the Amarna Letters, a collection of correspondence between Egyptian, Hittite, Assyrian and (possibly) Aegean rulers that forms the bulk of Cline's written sources. We also have a few shipwrecks, such as the one off Ulu Burun, which carried large amounts of high-value goods and might well have been the actual agents of this process.

Most current interpretations of the 'palace'-states of Bronze Age Greece see this exchange as the main lifeblood behind how they worked. On this, the defining article is Susan Sherratt's 2001 'Potemkin Palaces and Route-Based Economies'.1 Sherratt's image of a 'palace' is of an institution centred around a powerful individual, who gains access to prestige goods and eventually to these networks of long-distance exchange. Not just anyone can send a gift to the king of Hatti and expect something in return - the Amarna letters make clear that this is what Cyprian Broodbank has called 'a great-power club', where part of the point is that rulers only bother with people worth their time. He then distributes these goods among his loyal retainers, who become, through distributing lesser goods among their own followers, the fundamental core of his power and authority.

Three big things seem to happen towards the end of the LBA. Firstly, this all rests on exotic (especially Near Eastern) goods having a unique prestige value, and this seems to have been increasingly uncertain - much more work needs to be done here, but it seems that powerful people were no longer so concerned about certain 'traditional' expressions of prestige, particularly the age-old habit of depositing exotic finery in a monumental tomb. Secondly, the political troubles in Hatti, the Near East and Egypt mean that these goods are no longer coming - these networks are all about personal relationships and precedents, and have no durability when rulers and political setups change.

Finally, we have the big, technological change - essentially, the coming-together of a number of shipbuilding and navigational innovations that mean that ships can travel further, faster, carrying more stuff. This opens the door to much more 'private' commerce (and, as I mentioned in my first answer, private robbing and raiding), loosening the grip of the palaces on their prestige goods - if you can now get a fancy sword or a gold ring from the traders who pop into port every few months, suddenly there isn't the same motivation to devote yourself to a palatial overlord in return for the same things.

More importantly, these technological changes also change the map - now that ships can travel further over open sea, they are no longer stopping in the same places to repair and re-provision. While the goods coming from the Near East were vital to society in the Aegean, there's little evidence that it worked the other way around: we do see a few Aegean imports in e.g. Egypt and Syria, but by and large these people are interested in metals that come from further west. So there's a lot more movement of goods going on in the Mediterranean, but it's no longer coming as reliably to the Aegean palaces, and no longer in the way that palatial leaders had come to rely on to maintain their status. Indeed, if you look at the places that seem to have done as well or better in the Early Iron Age than they did in the LBA, they are almost all coastal places in a position to remain or become relevant on these faster East-West routes.

To put it simply, technological and political changes rewrote the rules of the political game: those political systems that had relied on them to justify themselves, mainly those on the Greek mainland, became irrelevant and had to change.

Sources

'Broodbank' is Middle Sea as referenced above. This is very much up his street - his early work is on island connectivity in the Early Cyclades, which itself provides a case study in what happens when ships get faster and no longer need to stop as often - the prominent sites of the Cyclades in the Early Bronze Age are minor if even present in the Late Bronze Age, because sailors are now simply bypassing them.

1 Susan Sherratt (2001) 'Potemkin Palaces and Route-Based Economies' in Economy and Politics in the Mycenaean Palace States, S. Voutsaki and J. Killen eds., pp. 214-238.