I agree with everybody that there's a very good chance it's just generic rejection language. That and "didn't connect with the story." On the plus side, you're getting personalized rejections, which means you're getting close
That said, to answer the question, what makes me connect with characters is usually little moments of weakness or humanity or longing. For whatever reason, readers like to want things along with your character, and often it's the little stuff, rather than the story's main arc or goal, that endears characters. Most people can't identify with a crazy premise like "being forced to fight in a televised death-match" but they totally latch on to the "need to protect your little sister" thing. That's the point where they connect with Katniss and are willing to follow her through three books.
This is something I'm trying to learn how to do myself, so I've been studying books I'm reading and pinpointing moments where I was like "yep, I'm sold on this character and want to know everything about her." Right now I'm reading The Bone Clocks, and it's got a zillion POV characters, all really well developed, so it's awesome for my purposes. Here are a couple of (not really spoilery) examples:
Character 1 is a whiny teenage girl. She's quite snarky and funny, but I'm pretty meh about her until I start learning details about visions she used to have as a kid, strange people appearing, weird stuff happening. Then I start to think "hmm, there's a lot more to this girl than she's letting on." Then she gets her heart broken and I really feel for her, especially the way she tries to hide her feelings behind a tough exterior, but still basically dissolves into a crying mess.
Character 2 is an unfeeling sociopath asshole. Total social climber, takes advantage of every single person in his life. Then he gives a homeless guy £40 for absolutely no reason. Suddenly I'm a little bit invested. Then he falls in love with this woman in one of the best-written falling-in-love scenes I've ever read and I honestly don't care if this guy murders a bunch of puppies next. I'm sold.
Character 3 is a war journalist who spends all his time traveling to dangerous countries, leaving his wife and daughter behind. I'm honestly a little bored by him at first, but the war stories are so fascinating that I can't stop reading them. Then he goes home and struggles with being a parent again to his daughter and she's busy testing his patience and being a normal kid, and he's guilty and confused and loves her so madly that I can't help but feel for him. Another character I've bought into.
OK, I've gone on way too long. Again, don't read too much into "didn't connect" rejections, but this is kind of a big thing for me lately so I figured I'd share.