The best way to think about them is basically investing in Mutual Funds wrapped in an insurance policy AND taking a sum of money and turning it into an income stream.
They can guarantee a stream of payments for a specific time (called a "Period Certain" annuity - as an example, the Powerball Lottery payout scheme is a period certain annuity - 29 years) or life. Some also offer a "Step Up" provision in which the withdrawal base is increased regardless of what the market as a whole does. This effectively allows for an annual increase in the payments made to you.
This quote:
I get it, there is NO flexibility. Once you sign on the dotted line, your cash is gone to XYZ Insurance Company.
is not entirely accurate. It is gone for as long as you want them to have it! BUT.....Almost all annuities come with a "Surrender Period". That means the Insurance company is going to charge you a percentage of the balance if you want your money back. If you want to
Surrender the annuity. It is always a sliding scale, starting year one with the highest percentage and going down from there. Some as short as 3 years, some as long as 12. Keep this in mind; The longer the surrender period, the better the commission for the broker.
There are fixed annuities and there are variable ones. Fixed are just that - the sum is fixed, the payout is fixed, etc. Low risk, low reward.
Variable annuities are just that - variable. Typically invested in a way that allows the balance to vary, both up and down in value by investing in securities that change in value, such as mutual funds. They can be very aggressive portfolios or very conservative ones. Most Variable Annuities will allow changes in investment portfolios. You can change it from conservative to aggressive along the way, it just depends on the issuing firm and the rules pertaining the the specific annuity.
It's good that your broker is skeptical. That shows he does have your interests in mind.
They can be an excellent place for a portion of your assets. Your age and other factors determine how large of a portion it should be. Typically the older you are, the safer it is to increase the portion, all the way to 100% if needed.
Here is an answer I gave on the old DU to a related question.
I hope that helps a little.
If you have more specific questions, just let me know.