How, in Buddha's name, did Tibet grow so large for a time?
First, this map is misleading : it accounts as part of the Empire places in the Tarim basin or India that were only sattelized, clientelized or vassalized. The empire didn't as much grew so big, than it was able to project its power and influence outside Tibet strictly speaking, as T'Ang China did, possibly succeeding it more or less bluntly.
I think, keeping in mind that I'm not that knowledgable about the Tibetan Empire, that it was a good exemple on what I tend to call "Borders Kingdoms", entities that arise in the peripheries of two or more political ensembles, arising from the crisis of these, but not wholly stable when they reinforce. The late Carolingian kingdoms of Burgundy/Provence, the abortive Burgundian state, Pampelune/Navarre, etc with various fortunes (Aragon did relatively well, State of the Paulicians did not).
The decline of T'Ang China after the An Lushan rebellion, which was particularily noticable in its western holdings and protectorates, as Turkic armies played an important role there : similarily to Tibet, the Uyghur Khaganate did took a fair share of these (and eventually more than Tibetans did). But I think that the fall of Gupta Empire as well beneficied the Tibetan build-up, in the replacement of T'Ang influence there, and as well the unability of Abassids to really get an hold in Central Asia.
Tibetan Empire more or less filled a geopolitical vaacum, at the cost of a political mobilisation and stress that, AFAIU, caused the civil wars of the late IXth century (altough maybe the Abassid economical crisis could have played a role with the decline of trade roads? I don't know : if someone did and coult correct or confirm, it would be nice).
How can we prevent, or at least delay, its collapse in the late 9th century?
It would be relatively hard : it's more a takeover and a succession in a geopolitical vaacum than an actual direct build-up : capacities of political and economic mobilisation in Tibet itself might have been limited, and dependent on the authority of Tibetan rulers over outer regions. The Tarim basin was eventually lost to Uyghurs and Kirghizs relatively early on, and the pressure in Central Asia (mostly modern Afghanistan) from Abassids was a bit too much : most of sattelization declined in face of the unability of the empire to really hold its own IMO.
I think the best chance might be a Tibetan conquest of southern or central China, at the cost making it less a Tibetan empire, but at least a more Sinicized empire of Tibet, if not effectively a Chinese dynasty.