So, first of all, the requirement is that you have made $200K in each of the two most recent years and expect to make as much this year. Or, $300K if you combine your spouse's income. Or have a net worth of over $1M. (There are other alternatives, but those are the main ones for individuals.) Those requirements come from Regulation D under the securities laws, which define what it means to be an 'accredited investor.'
Why is that important? Because the company has to give non-accredited investors a bunch of information, including things like an audited balance statement. That's all an expensive proposition for companies, so it's much easier for them to deal only with accredited investors.
Those restrictions came about because the SEC figures that accredited investors are capable of protecting themselves in private offerings while unaccredited investors aren't as sophisticated.
Note that an "Angel Investor" does not necessarily have to be accredited -- it's just that most startups won't take money from unaccredited investors.
Also, Congress just passed the JOBS act, which the President is expected to sign. This act will allow "crowdfunding" by unaccredited investors, so you may yet have your chance to invest.