Agency Mortgage Backed Securites are the pools of loans packaged together by Fannie Mae and Freddie Mac to be sold as investments. Once the loans are packaged together and sold they resemble any other typical bond-like instrument. This means that they have both a price and a yield.

Price is the cost of buying that bond. Yield is the return on investment that you receive from the investment. Think of it this way, Price is what you pay to own the investment, Yield is the interest you receive, like a dividend, because you own the investment.

As to how the Federal Reserve, or anyone for that matter, buying MBS benefits the economy, here goes:

Just like a bond, the higher the price, the lower the yield. Think of it as a teeter-totter, if one side is high, the other side must be low. Now think of typical supply and demand. If demand is high, prices rise. If many people want what you have, you can sell it for a higher amount. So, as the Fed buys MBS, prices rise. Now go back to the teeter-totter reference, if prices rise, yields (interest rates) fall.

Many economists, including leaders of the Fed and the Treasury has surmised that stabilizing home prices is the best way to slow or stop the current economic issues. However, they cannot arbitrarily stabilize home prices. The best thing that they can do, is to work to lower interest rates. By lowering rates, the thinking goes, buyers who are on the sideline will purchase homes. Going back to a supply and demand principal, once the demand equals supply, prices stabilize. Therefore, if the Fed, via lower interest rates can entice enough buyers to purchase homes, at some point home prices stabilize. Stabilizing home prices will have a myriad effect on many areas of the economy. It will help those directly involved in Real Estate, as well as ancillary business such as home builders, developers, etc. Home values are also considered a good indicator of the economy as a whole, so stable values will have a trickle down effect into consumer confidence as well as other unrelated areas of the economy. In the end, the Fed buying MBS to lower rates is not a cure all, but it appears to be a very substantial step towards trying to move the general economy forward.

The price of mortgage backed securities (MBS), like all commodities, is determined by supply and demand. When the Fed buys MBS the supply decreases and the demand increases, therefore the price of the MBS goes up. The yield on MBS (bonds) is inversely related to the cost. I.e., when the price goes up the yield or interest rate goes down.

So the Fed buying MBS should lower the interest rate on mortgages. This lower cost of financing homes has two effects. One, more people will be inclined to purchase a home. Here is supply and demand at work again. More demand for housing would lower the supply. If more people want to buy homes due to lower interest rates, the prices of homes would increase or at least the rate of decrease might slow. 

The second effect is if consumers that own homes are able to refinance and lower their interest rate they have more money in their budget each month to spend on other things. This increased consumer spending would help stimulate the economy.