I think you almost answered your own question. You are almost correct when you said 'United States Treasuries either buys or sells from 18 primary dealers'. United States Treasuries are either bought or sold by the Federal Reserve from 18 primary dealers. Broken down: Treasuries are being transacted between the Fed. and 18 others.

rule of Proximity [The Federal Reserve has 18 primary dealers. The Federal Reserve buys Treasuries from these dealers or sells Treasuries to these dealers.]

They're saying the Federal Reserve buys US Treasuries FROM 18 primary dealers. So there are 18 dealers FROM WHICH, the Federal Reserve buys said treasuries. What I'm sure is confusing you is the word "Treasuries," because you're probably thinking of the single institution known as the "United States Treasury," which is where the US government's revenue is managed, among other things. US TreasurIES (plural), are, to put it (almost) simply, when you invest money in the government through bonds, notes, or bills that are issued by the US TreasurY (governmental department), and then the government pays you back with interest, depending on the type of loan you chose (these loans are called "securities"...in fact, US Treasuries are basically securities through the US Treasury). An example would be the war bonds that the American people bought to help finance World War II. These Treasuries usually regarded as the safest investments there are, as returns are consistent, but not as rewarding as riskier investments. Therefore, the sentence is saying that the Federal Reserve, which is in charge of regulating banks and monetary value/policy in America, has 18 dealers (there are actually 21 now) for their transactions involving Treasuries--the "dealers" being, in particular, big banks and broker-dealers that turn around and buy/sell these securities to/from each other, such as Deutsche Bank in Germany, Nomura in Japan, or Goldman Sachs here in the US). Basically, this is all done in the effort to borrow money when the government needs to raise cash, and thus stimulate the economy if it happens to do something right with that cash. I hope that shed some light. I'm not very well-read in this area, and I might have gotten it wrong, but hopefully now you know what the sentence is saying, even if you don't know all the details :)