How does the NY Fed's trading desk use this process as a tool to adjust bond prices?
Investors other than banks (especially Money Market Mutual Funds and some GSE's (Government Sponsored Enterprises)) have cash they want to invest. The Desk offers reverse repos at an attractive rate, so these investors put their cash to work by lending it to the Fed, receiving government securities as collateral in return. ( As everyone knows the Fed has plenty of government securities on hand these days, over 2000 billion worth!). The market for short term finance is competitive so if a company (say General Electric) wants to raise money from MMF by issuing Commercial Paper they will have to offer a higher interest rate than heretofore, so other interest rates (in the case the CP rate) rise also. As we all know a higher yield on CP goes with a lower price for CP, that's just Finance 101.
So the NYFRB Desk is basically competing with securities issuers in raising short term money. In doing this they affect they yields/prices of short term investments and short term securities.
Reverse repo is a process whereby the FED sells their securities in the money market where only some institutions (e.g. big banks money market mutual funds) are able to purchase the securities off from FED.
FED sells their securities to remove liquidity in the money system, by reducing the amount of money supply, people will require to pay higher interest rate to get the lesser money supply from bank and other financial institutions.
Bonds are eventually debt securities, which pays out interest yielding. Treat it as an instrument form out of loans, therefore when interest rate hike due to lesser money supply, the bond interest rate (borrowing cost) goes up.
To decrease the increase rate, FED can purchase repo by purchasing securities from banks, which increases the liability and assets of the bank; thereby creating more liquidity through more money supply. When the supply is higher, banks will want to lend out by, which can be done by lending at a lower interest rate.