Not all stories can be successfully told in a linear fashion. If nothing interesting is happening, time skips in anime can help avoid plodding through meaningless filler, and jump viewers right to the interesting part. The best anime time skips lend the series a laser focus on what matters to the plot and the characters and don't dwell on the parts that aren't important.
Few people want to see what happens while most of the cast of Fairy Tail is in stasis - they want to see what they do when they're up and about. Few people want to watch Taki and Mitsuha grow up and apply for jobs in Tokyo - they want to watch them find each other again. However, sometimes time skips leave out important information - what kind of training did Jiraiya do with Naruto for three years, and why did he come out of it not knowing anything except rasengan? What exactly has Sasuke been doing with Orochimaru, and how badly traumatized is he by it?
Some time skips work, and some don't - it's up to you to decide which is which.
Makoto Shinkai's hit movie Your Name features a lot of time travel, but it also contains a time skip. At the end of the movie, after the residents of Itomori are saved from death by comet, and Taki and Mitsuha's respective timelines combine into one, the two of them stop switching bodies and proceed with their lives. Taki finds a job in Tokyo, and Mitsuha moves there and finds a job of her own. The process isn't detailed - instead, the two of them appear in Tokyo, six years older than they were, and run into each other by chance.
Naruto, Naruto Shippuden, and Boruto all take place along the same timeline, but each time a new series begins, there's a time skip.
Three years pass between Naruto and Naruto Shippuden, bringing the protagonist and his same-aged peers from age 12 to age 15. During that time, Naruto trains under Jiraiya - improving the rasengan technique and his taijutsu skills. Meanwhile, Sakura is training under Tsunade, improving her physical strength and learning medical ninjutsu, and Sasuke is learning forbidden techniques under Orochimaru.
Between the end of Shippuden and the beginning of Boruto, there's another time skip, this one of an indeterminate time period of at least fifteen years. During that time, Naruto becomes hokage, marries Hinata, and has two children. The rest of the cast changes in similar ways, becoming parents and establishing careers.
In Assassination Classroom, there's a time skip that shows viewers what class 3-E is like after seven years after the series, which takes place during their final year of middle school. Everyone's lives have changed dramatically, but here are some highlights. Nagisa becomes a teacher in a difficult school and uses the confidence he learned through Koro-sensei to earn his students' respect, and hopefully inspire them the way his former teacher did him. Karasuma and Irina, two special agents sent in to help train the kids and help assassinate Koro-sensei, get married. Karma gets a job at the Economics, Trade, and Industry, and Principal Asano gets fired.
In the final episode, the newly minted adults reflect on their experiences as junior assassins, and how far they've come since then
Tokyo Ghoul:Re takes place two years after the original series. During those two years, the war between Aogiri Tree, the CCG, and other organizations continues to rage. Most importantly, however, is a drastic change in the protagonist. No longer using his former identity as Ken Kaneki, he now goes by the name Haise Sasaki, and is leading a group of other humans who have gained ghoul-like abilities through surgery, and are working for the CCG to take the rest of the ghouls out.