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.
Just before the One Piece time skip, Luffy watches his older brother Ace die in order to save his life. Wanting to make sure that no one else he cares about has to suffer such a violent end, he decides that he and the rest of the Straw Hats should separate and undergo a two year period of training before taking on the dangers of the New World. The training process isn't depicted - instead, the series picks up two years later after it's complete.
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 order to protect Tenrou Island from Acnologia's devastating attack, Lucy and Mavis cast a powerful magic called the Fairy Sphere. This sphere offers amazing defense capabilities, but it also puts everything inside it into stasis. Thus begins a seven year time skip in which nothing happens for most of the main cast, but the world around them changes. Some viewers complain that they don't get any significant power-ups during that time, which happens in other anime with time skips - while others defend this on the grounds that a power up would be impossible given the stasis premise.
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