SPI_push should be called before executing another procedure that might itself wish to use SPI. After SPI_push, SPI is no longer in a "connected" state, and SPI function calls will be rejected unless a fresh SPI_connect is done. This ensures a clean separation between your procedure's SPI state and that of another procedure you call. After the other procedure returns, call SPI_pop to restore access to your own SPI state.
Note that SPI_execute and related functions automatically do the equivalent of SPI_push before passing control back to the SQL execution engine, so it is not necessary for you to worry about this when using those functions. Only when you are directly calling arbitrary code that might contain SPI_connect calls do you need to issue SPI_push and SPI_pop.