TLDR: Massive expansion of credit fuelled by rehypothecation, a general shift to repo, then the scale tips and everyone pays as credit collapses. Quants were there, but I don't think they can be simply blamed for all the ills of the world.
There is a general disagreement about what caused what, so some of this is guesswork. I'm marking this a community wiki so that other users can correct my mistakes, and so I don't get downvoted to oblivion for saying something stupid. This is just a starting point.
In the 90s there was a boom - rates were high, credit risks were lowish and demand was high for high-return investments to beat inflation and bond returns. In order to facilitate lending in this environment, quants began to rehypothecate assets with lower ratings into higher-rated assets via tranching; by combining a portfolio of variable quality assets into a vehicle that defaults on one tranche first, then the next, you can reserve the best tranche as a high quality asset. These now-acceptable high-rated assets could be used for repo (repurchase) collateral, and permitted a vast expansion of the repo markets as the pool of available capital exploded.
The rehypothecation process expanded into mortgages; by bundling and tranching residential mortgages, another vast pool of variable-quality assets could yield highly-rated collateral.
But this usage of repo brought with it another change; quants were increasingly using credit instruments to pin down and insure against counterparty risk. This increased awareness of counterparty risk as part of trading pushed interbank lending in the 00s away from cash-based lending and toward collateralisation. Yet the market still behaved as though 3m cash rates were a good indicator of funding costs, and specifically the Libor/Euribor indices were accurate measures of those rates.
This sets the stage, then; a large repo-funded expansion of credit, where the repo collateral is largely rehypothecated debt based on commercial paper, mortgages, bonds and anything else that could be bought and tranched. Because of all this available credit, and the high value placed on any asset that could be sold, lenders expanded into subprime mortgages, among other things, and standards for borrowers slipped. Many businesses by this point had opted to sell and lease back assets or run on low cash margins as overnight and short-term lending was easy to obtain and enabled more aggressive business practices.
Where are the quants? Sitting happily with complex models which assured them that something very unusual would have to happen for their hedges to go wrong, or their counterparty protection to dry up.
The trigger came when an increasing number of those subprime borrowers began to default; banking had undergone a transition from an austere and risk-averse business to a highly retail-oriented one, and mortgages had been offered with low introductory rates. When borrowers transitioned to the higher rates after introductory periods, they began to default.
What followed was an avalanche of downgrades; first subprime mortgages were being downgraded. That, itself, wasn't too much of a problem - the number and proportion of subprime mortgages was not huge. But what it did do was drag down the ratings of mortgage-backed assets. The repo system is based on the ratings of collateral, not the actual type, so as this class of assets dropped below required grades, institutions had to start withdrawing other asset classes to replace them. But this pulled the rug from under some institutions who were overexposed, and their ratings started to fall. The greatly-expanded pool of repo collateral and thus created money began a sudden contraction.
As the pace of this contraction accelerated, a lot of quant models started to fall apart. The size of the movements was outside the scope of normal or lognormal models, and the interdependency of risks was suddenly realised. An unfortunate position where much of the insurance against counterparty risk was backed by a small pool of underwriters like AIG meant that not only had counterparty risk suddenly spiked, but the value of the insurance dropped.
Credit contracted ever more quickly, taking company after company and institution after institution with it. But the remaining players held OTC instruments with other counterparties and there was no transparency regarding exposure. All remaining un-collateralised lending dried up, because a bank could not tell who had now-toxic positions and the chances were that whoever wanted to borrow was a liability. The credit contraction, caused company failures and then job losses, which caused the economic outlook to worsen, and the rest of the market (outside credit and money) turned to a bear market.
In the natural sciences, the laws of the universe are unchanging and impervious to the experiments you do on them. Momentum appears to be universally conserved. In finance, however, all the models are based on what the quants can see in the market, and the market is always changing. The boom lasted so long that the market fundamentally changed its construction - unsecured lending was no longer the normal way to do business, which had made collateral itself the keystone of liquidity. The models, which had been back-tested and so on, were not built to handle all of the above occurring together; there were quotes like the amazing one from Viniar, CFO of Goldman: “We were seeing things that were 25-standard deviation moves, several days in a row,” - if you're seeing 25 standard deviations at all, your model is broken.
It is possible to blame quants for a lot of stages here; for misunderstanding the risks, for creating the world of rehypothecation, for missing the correlations between risks, etc. But remember that beneath the bubbles were trends; the internet and electronic communication were spreading across the globe, business practices were changing, the sophistication of financial models was blossoming. Inside a bubble, everything feels like a trend; the shift to collateralised lending must have felt, if anything, like a reduction in risk rather than an increase, particularly if you compared a portfolio based on unsecured lending with minimal counterparty protection with one fully collateralised with sophisticated hedges against various counterparty risks.