Unlocking Job Breakthrough is important in game because it's one of the requirements for Third Job Class like Warlock, Rune Knight, Royal Guard, Shadow Chaser for incoming game episodes. Currectly the the max job level is 40/40 and unlocking Job Breakthrough can extend your Job Level to 70/70.
Job Breakthrough also unlocked more powerful skill levels, for example Sonic Blow of Assassin which has 10 levels only, Job Breakthrough can extend to 5 more levels which deals more damage and stun effect. For more info visit Job Class and Skills.
Before continuing, be sure you have the following.
Once you reach job 40/40 of your Transcended Job, a quest will automatically appear on your quest list. To proceed, find Sjilly, near the Dummy Training inside Prontera. She will as you some set of questions after than she will then ask you to proceed to Higgy Eez for further requirements.
Higgy will ask you to surrender the 30 Silver Medals and pay 500,000 zenny as payment for extending your jobs. After that, your job will be extended to job 43.
Extending your job level, you need to acquire Peak Shards and this Peak Shards can only be bought inside Guild Hall vending machine. You need 9 Peak Shards to add 3 more jobs.
To extend, find Alice Fona in the guild hall to exchange your Peaks Shard for 3 Job levels.