The two-sentence story is that as an investment bank, Lehman could at any time rapidly invest in a pile of crap, and so lenders would only extend it short-term credit. Therefore any impediment to Lehman's constant rolling over of debt would kill it. Extensive quote below, emphasis added.
Lehman’s demise came when it could not even keep borrowing. Lehman was rolling over at least $100 billion a month to finance its investments in real estate, bonds, stocks, and financial assets. When it is hard for lenders to monitor their investments and borrowers can rapidly change the risk on their balance sheets, lenders opt for short-term lending. Compared to legal or other channels, their threat to refuse to roll over funding is the most effective option to keep the borrower in line.
This was especially relevant for Lehman, because as an investment bank, it could transform its risk characteristics very easily by using derivatives and by churning its trading portfolio. So for Lehman (and all investment banks), the short-term financing is not an accident; it is inevitable.
Why did the financing dry up? For months, short-sellers were convinced that Lehman’s real-estate losses were bigger than it had acknowledged. As more bad news about the real estate market emerged, including the losses at Freddie Mac and Fannie Mae, this view spread.
Lehman’s costs of borrowing rose and its share price fell. With an impending downgrade to its credit rating looming, legal restrictions were going to prevent certain firms from continuing to lend to Lehman. Other counterparties that might have been able to lend, even if Lehman’s credit rating was impaired, simply decided that the chance of default in the near future was too high, partly because they feared that future credit conditions would get even tighter and force Lehman and others to default at that time.