If you google the whole sentence (in quotes) "I felt a real sense of belonging" you can see some great examples of when people use this phrase.
For example, one person described the first time they attended a new church. Even though the people there were strangers, they welcomed him and were kind to him. He immediately felt like he was a part of the group and felt a sense of belonging.
Another person used the phrase to describe a job interview. Everyone she met at the company was warm and friendly, and she agreed with their ideas and liked the way the company was run. She felt that she would enjoy working at the company and feel comfortable there. Again, "I felt a real sense of belonging."
I hope this helps!