How does money in an IRA grow?

Im confused as to how say 2000 dollars a year in an IRA will grow to be more money later on in life if I am only making 2000 dollars investment every year. It seems as if I'm just opening another savings account

An IRA is just a term that describes the body of laws the underlying investments are subject to. Think of the IRA as an umbrella, holding your investment types underneath. It sounds like your IRA is just holding a core account that is likely a money market account, so, yes, that acts similar in nature to a savings account. But within your IRA, you can allocate that money to stocks, bonds, mutual funds, and all sorts of other investments instead, which (generally) make much higher returns.

I suggest exploring ways to diversify your IRA holdings into more long-term growth oriented investments like equity funds and targeted-date mutual funds (which might be best for someone who has little interest in tracking his/her fund on a regular basis). Depending upon which financial company manages your IRA, it should be easy to call them up and transfer that money (within the IRA, not out of) to a more lucrative investment type that will yield you better returns over time.

It is a savings account. Compound interest is how your few dollars, over a long period of time, grow to many dollars. Say your $2000 earns 10% the first year. At the end of that year you now have $2200 dollars($2000 + 10% of $2000($200)). The next year you would have $2420($2200+10% of $2200). It continues to grow that way for say 20 or 30 years. Now that alone will not make you rich. But, with a IRA that you contribute to many times a year your principle grows much faster. Each time you earn interest your principle grows in addition to regular contributions. The combination of all that equals a very tidy sum at the end of say 20 or 30 years.
I know that seems like a long time, but the last trick is to have your contributions to a retirement account taken out of your income before you get your paycheck. Then you live on your take home pay. Like most people, you live on your take home pay and don't consider that invested money because you don't get it in your hot little hands directly. Another component is that if the IRA is at a workplace often companies will match the first 3-6 percent of what you contribute. This is like free money. If they match dollar for dollar that is a 100% return on your money right off the bat. Look up a online calculator for compound interest or retirement accounts. You will see how money grows over long periods of time.

Within that account, you invest the money you deposit every year.. so you can buy and sell stock to make money, or buy stock and receive the dividends, or buy cd's with it which receive interest, or invest it into mutual funds which will hopefully increase in value.. all within that account.
If you ever work for an employer who offers a 401K take it! It's free money. You can have that and the ira.

It is called "compounding" and "reinvesting" of your money. The money you gain on each investment is put back into the IRA account. Then THAT money also earns a return ... and so forth.

It is "just another savings account", except that you don't have to pay taxes on the interest or gains until after it has finished growing (your retirement)

Just as with any savings account, $2000 that grows for 30 years will be worth a whole lot more than $2000 that grows for only 15 years.