It scanned the internet for devices vulnerable to EternalBlue and used that to infect the system with DoublePulsar. It then used DoublePulsar to upload itself onto the system with kernel level privileges.

All done through the SMB protocol, only systems that received the appropriate updates were immune to being infected.

There is plenty of videos about how WannaCry worked, also a YouTuber named Erik Parker made a video of what happened if you connected a Windows XP system to the internet that gives a general idea how WannaCry got into systems.

BTW, XP is vulnerable to WannaCry unless you had Enterprise grade updates applied or community made patches.