First the South Sea Company wasn't truly an independent company, it was more a quasi-company built by members of the government to service its debt. The British government was massively in debt both internally and internationally, so initially you could swap your debt for shares in the South Sea Company (at a lower rate of return) but with the possibility of the shares being worth a lot more.

For why the British were taken in? For a start the ability to find out information about what the company was actually doing was limited as there was no legal requirement to file accounts and the problems were deliberately kept hidden. Secondly the bank engaged in a massive advertising campaign brining in the royal family, authors and other respected people to make it attractive, if the king is buying into it, its must be good. Finally the South Sea Company styled itself as the next East India Company which was making massive profits but wasn't selling shares.

Here's a nice video series about if your interested: https://www.youtube.com/watch?v=k1kndKWJKB8