Hi, Christian,
If you use SWO, you have to use SWD protocol(includinf SWDIO, SWCLK, SWO(optional)), the SWO pin of LPC4088 must be connected in the onboard CMSIS-DAP LPC-Link2 debugger and an external SEGGER J-Link probe.
secondly, you have to use PRINTF() function in application code so that message in PRINT() function can output via SWO.
I attach a doc which demos how to use SWO.
Regarding the question where the PRINTF() function will output, it is dependent on the tools, the PRINT() function can output to console, uart, ITM.
Hope it can help you
BR
XiangJun Rong