Disclaimer: I'm not an expert on the political aspect of the crash-hopefully someone with knowledge of British politics in this time would be able to contribute here.

Simply put, the British government had a substantial amount of debt due to large amounts of government deficit spending (primarily military). What happened was that certain persons created a joint-stock company called the South Sea Company. The idea was that this company would have the exclusive right to trade in the "South Seas," or South America (which incidentally was entirely controlled by Spain). The government offered debtholders the right to convert their debt into stock in this company. What that means is that, if I have a 10 year government bond, instead of being paid interest for 10 years and then getting my initial loan (or principal) back, I can instead exchange the bond for shares in the South Sea Company (so I would trade a bond worth $1000 for $1000 worth of stock). As an investor, I may want to do this because a) selling stock is easier than a bond (the financial term being "liquidity," or how easy it is to turn something into "liquid" assets like cash) and b) I might think I would get more return for my investment (as stocks, or equity, typically appreciate in value, whereas bonds remain at a fixed rate of return).

However, this doesn't mean anything if the company doesn't have any actual assets. What the government did was they took government debt equal to the amount exchanged and "borrowed it" against the South Sea Company. This essentially allowed the government to pay the debt at a lower rate of interest and at a more convenient schedule. At the same time, the South Sea Company now had assets in the form of government debt, so an investor would be comfortable knowing that if the company collapsed, he would at least have a claim on monies owed by the government-which was by far one of the more stable financial institutions. So it seemed to be win-win for me. However, this was a bit of an illusion.

Essentially, the company's stock began to increase in value as speculators thought that the South Sea Company had a thing going (with the monopoly on South American trade that didn't really exist) and with the comfort that if it turned out to be naught, at least there would be government debt behind it. However, they then would make take loans using this stock as collateral. Essentially, they would go to a bank and say, I want to take a loan to do XYZ. I will put up stock worth something as collateral to mitigate the bank's risk so you will want to loan me the money.

Eventually the price reaches a certain amount (iirc it was around ~1000 GBP). People start selling their stock to make a profit and the price starts going down. Suddenly banks who had loaned money against the stock as collateral have a problem. The collateral isn't worth as much as it should be, so they demand that the people who borrowed the money either add more collateral or repay the loan immediately. If the loan cannot be repaid (which was usually the case) the bank would foreclose on the collateral (seizing the collateral, which is the stock) and try to liquidate it (sell it). This in turn drives the price of the stock down even more, triggering other lenders. This "trigger" is known as a margin call. As a result, many people went bankrupt (as they couldn't pay the loans) and investors were ruined as the stock price collapsed. Needless to say, a lot of people were very angry at losing a lot of money.