I am very enthusiastic about Bitcoin Unlimited and think it is a great idea. (Disclaimer: I am a big blocker, and see BU as the mechanism that will end up bringing higher maximum block sizes that can let Bitcoin scale with on-chain transactions. Anyway, my stance on big blocks is, I think, independent of my argument below.)

As Zangelbert Bingledack has argued before, BU lets users do something that they can already do, albeit currently in a more complicated way. Let's suppose a group of miners decide that a blockchain with 2 MB blocks is acceptable to them, and want to gamble that other miners and economic actors (like exchanges) will think likewise and accept such big blocks too. They can already upgrade to a blockchain that supports larger blocks by recompiling their own custom version of Bitoin Core where they substitute a higher value for the integer constant MAX_BLOCK_SIZE. If these miners command a minority of the total hashing power, they will find that whenever they try to relay a 1.9-MB block, it gets stalled among nodes that refuse to relay it and the block gets orphaned and rejected by the longest chain. Now they could keep on trying to propagate their blocks and use human-communication channels like this forum to try to attract other miners and users to the 2-MB camp, but there is something that will probably put them off and that makes it hard for this approach to succeed: the moral aspect. Right now, the more common view is that a block that exceeds the 1-MB cap is an "invalid" block that violates the consensus rules, just as would be a block that contains, say, invalid transactions or incorrect hash values. This leads to a situation where the miners sending 1.9-MB blocks out to the open will get rejected as "malicious" or "attacking" nodes. Any reputable mining company won't want to be associated with such antics, and will understandably keep their blocks under the limit to be seen as well-behaved legit nodes.

Now the great idea behind BU is that it completely eliminates this moral aspect by taking the maximum block size out of the consensus rules. Under BU, the blockchain becomes a parameterised family of N-capped blockchains. If BU becomes the main Bitcoin implementation, miners may freely choose to decide that they can support a 2-MB-capped blockchain or a 32-MB-capped blockchain without anyone shouting at them "malicious!", "attacker!", "you filthy rascal, impostor of a node, boo, boo, boo!!!". The moral aspect vanishes as it is now just as legit to adhere to any cap. Those who only support the 1-MB-capped blockchain will reject any blocks bigger than 1 MB, while the 2-MB-capped blockchain nodes may keep on trying to build a longer chain. It's important to note that the blockchain only splits when a subset of the miners advocating for an N-capped blockchain command more hashing power than those miners that advocate for M-capped blockchains such that M < N. At that point, two things could happen: a node supporting the M-capped blockchains may accept that a longer chain now has larger blocks and raise their M parameter to match at least N, the new proof-of-work consensus, or insist that they won't support such large blocks and keep on mining a shorter chain of blocks that respect their M limit. This might lead to several alternative chains coexisting for some time, but the ensuing chaos should be very brief as the free market will rapidly settle for one of them. My feeling is that this would be the chain with the higher block sizes and the larger proof-of-work behind it.

How will the growth of the block size cap happen under this mechanism? I think the pressure on miners to increase the limit will grow as the cap starts to be hit. As more users try to send (on-chain) Bitcoin payments, the 1 MB limit will be hit more and more often and a backlog of transactions will start to grow. This will lead to users complaining about their transactions being stuck in limbo and some negative media reporting about it ("Bitcoin network collapsing" and the like). Under such strained conditions, which we may start to see pretty soon, the Bitcoin price will take a battering in the exchanges with miners starting to worry about their income and realising that they're jeopardising the Bitcoin network and their earnings by not raising the cap. I could be wrong about all this, of course, but then it would be the free market which would prove me wrong by settling on a tightly capped blockchain. In any case, thanks to the mechanism that BU provides, raising the block size limit can be a market-driven smooth transition rather than an incessant and unseemly fracas among the core developers in the mailing lists and IRC channels.