FATEs are located in several areas, and show up on your mini map as a purple icon with a blue circle around them. When you enter the blue circle, you join the FATE and help everyone to achieve a common goal. Stay inside the blue circle until the FATE ends and the trophy will unlock.
Please note, depending on your level vs the level of the FATE you may see an option pop up that asks you to Level Sync. Be sure to click on that option so the FATE will count by automatically adjusting your level as needed.
You can also get excellent experience by joining FATE groups. You may see people advertising in chat for "LFM FATE Group!" if you send them a /tell they will invite you and you can follow the group from FATE to FATE racking up combo kills and big bonus exp rewards. This helps the grind to 1000 fates go by much faster.
Dont forget to eat or drink a buff item that not only increases stats but gives bonus exp, usually for 30 minutes. You can buy stacks of 99 of these from vendors in town. Look for whatever food or drink compliments your primary stats for best results.
Do this 999 more times to unlock: