This is a nice board, but it does not have the Mbed HDK or debug probe hardware on board. This is necessary for the drag & drop functionality. The LPC1768 has a bootloader in ROM, but you need some flash tool for programming.
The mbed LPC1768 board is, in addition to the main LPC1768 MCU, equipped with another microcontroller denoted as IC6 on page 3 of the schematics. The firmware for this auxiliary processor is stored in the AT45DB161D flash memory (see page 1 on the schematics). As I know NXP never disclosed what this additional microcontroller is.
After checking the schematic of your board I did not find any additional processor neither an AT45DB161D flash memory. So I’m afraid there is no firmware running on your board to be updated .
That’s a good question. Since NXP, in order to prevent cloning of the Mbed LPC1768 board, did not disclose any information about IC6 the only feasible way seems an external DAPLink programming/debugging board. I’ve already purchased this one on Ali Express but had no luck to make it work on Linux (I don’t use MS Windows). It doesn’t seem to support drag&drop programming but should enable debugging.
However, there are available other DAPLink interfaces enabling drag&drop programming.