What are the rules for a Thai 60-day multi-entry tourist visa and extensions?

How did you apply for this? Do you have to go to a consulate? I think there is one in Boston.

"approved for a Thai 60 multi entry" I am not exactly sure what that means.

However, a Multiple Entry Tourist Visa (cost +/- $200) allows you to stay up to 9 months but you will need to exit every 90 days. If you return (exit and return) on the last day the visa is valid you can stamp in for 90 days which means 6 months + 90 days.

Let me explain the 90 days. Each time you enter Thailand you will get a stamp in your passport. That stamp will include "Permitted to stay until...". Depending on your visa situation the stamp can be valid from 15 - 60 days. In your case it's 60 days. Near the end of the 60 days you will have to decide to, extend or exit. You can extend 30 days or you can exit and return (same day) and get another stamp for 60 more days.

To extend you will go to the local immigration office, pay 1900 baht and get a 30 day extension. In order to do this, you must have a valid TM30 on file. If you are staying at a reputable hotel they will do the TM30 for you. If you are staying with a friend, AirBnB, or other smaller hotel, they will likely not do the TM30, or at a minimum you will need to verify it's been done. If it has not been done you will need to post another question about that.

You can get almost 9 months if you do two border bounces

Yes, everytime you enter Thailand up to the Visa expiry date you get stamped in for 60 days and every entry stamp can be extended for 30 days if required for 1900B giving you almost 9 months in Thailand with careful planning.

The visa is valid for 6 months from the date it's issued. If you look at the visa it should have a final date on it.

Every time you enter Thailand until that date, you'll receive a stamp for 60 days. You can apply once per entry for a 30 day extension at your local immigration office.

If you leave and return just before the visa ends, you can get a final 60+30 as well.