1. Such characters show me something about my own nature (for good or bad!).
2. I can identify with the characters, sympathise even.
3. I will "root for" characters and "feel" their struggles, which is not quite the same as 2 above. For this, I have to really like the characters concerned. With 2, I can identify with say what a villain is up to (they've been crossed once too often and are now out for revenge), but I'm not going to root for them to succeed in their aims. I often hope ambiguous characters will not turn out to be villainous in the end or at least have motivations that are understandable. The best of these will do both.
4. You can "see" exactly where a character is coming from. That leads to empathy (which I believe can encourage empathy generally and that is no bad thing).
5. Characters will show you the world they live in and how they handle it. Is there something I can learn here? (That includes what not to do!).
6. For a character that's set in a historical period, you can compare how they handle their situation and ponder how you would do so. (I love Elizabeth Bennett in Pride and Prejudice but can see her situation as almost being prison like. For her not to marry Mr Collins, when it would have meant helping her family, was a brave thing to do. It would also be seen as selfish, especially by her mother. Here, I am so glad I have never faced something like that).
7. As a writer, you can put your characters through the emotional wringer! Heartless though that may sound, it is also huge fun - and it will be where your story really is. It is all in the conflicts.
8. Sometimes a historical character can change your mind about a period in time or a well known historical figure. The Daughter of Time by Josephine Tey did this for me with regard to Richard III and Henry VII.
9. Characters can show up injustice clearly. Think To Kill a Mockingbird here or something like Uncle Tom's Cabin.
10. Characters can say things you would love to say to others but for whatever reason rightly decide it's best not to! And I'm not saying more than that....!