The $60 plan is a pay by the month plan. This means that once a month they take $60 out of your account and you then have service for the month. In this case, he is saying he puts the $60 in every month just a day or so before AT&T takes the payment for his monthly account, so for 29 days or so each month his account shows a $0 balance.
It seems you are confusing monthly plans with pay as you go plans. With monthly plans they pull the full cost of the plan out of your account when the previous month expires, in return you get an allotment of minutes, texts and data -- and you can use those until you run out of minutes/text/data or until your month ends. If you run out of minutes/texts/data, typically you can pay for extra minutes/texts/data or you can restart your month (in which case you have until that day on the next month before they pull another payment).
With pay as you go, you put money into your account but they only take your money as you make calls/texts/or use data. For example, I have a Lycamobile plan as a backup (I have a T-Mobile $30/monthly plan that is my main mobile phone), and Lycamobile charges me 2 cents/minute for talk, 4 cents per text, and 6 cents per MB of data. In this case, I have just over $2 in my account and if I make a 5 minute call they will deduct $.10 from my balance. And, as you mentioned, if I don't use my phone for a few months, or don't add money, then my $2 will expire.
So, the difference is, with pay as you go you actually pay for what you use. With monthly plans they take a set amount of money each month and they give you a bucket of talk/text/data that you can use. With monthly, even if you don't use your phone they still take that full amount of your monthly payment -- as if you used the full amount of talk/texts/data that the plan gives you.
Also, please realize I am talking prepaid plans here. Postpaid plans work similarly to the prepaid monthly plans except that you pay after your month has ended. That means that if you use more than what your plan allows that you can get a very large bill at the end of the month for the extra talk/text/data you used. With prepaid, you pay first and, if there isn't enough money in the account, rather than billing you they cut off your service until you add money to your account.