What does it mean that the boss told me to punch in only for half a month and not for the second half?

it means that you only need to punch in the first half of the month.

the boss told me to punch in for half a month, and not to punch in the second half of the month means that you only need to punch in for half a month.

Punching in refers to putting the magnetic card on the magnetic card machine so that it can read the relevant contents, especially when the staff inserts the attendance card into the magnetic card machine to record the time of arrival or departure from the unit when going to and from work, which can be understood as "signing in, registering, showing certificates" or "paying the bill, please" and so on. "Card" is a very common thing in modern society, such as credit card, shopping card, bus card, bank card, membership card, etc., followed by the saying of "punching in", which has a smooth pronunciation and concise semantics.