For something 'to belong somewhere' indicates that it is in the correct or right place. To say that something 'belongs' may or may not suggest what is called a 'value judgement' (indicating that something is 'good' or 'bad'). For example, we could say 'a dolphin belongs in the ocean'. That is to say, 'a dolphin does not belong in the zoo'. 

A 'sense of belonging' is important for human beings, and this phrase is quite commonly used.

The 'sense of belonging' is the feeling of wellbeing we derive from feeling that we 'belong' in a community (i.e. that we are right to be in the community; that we are a member of the community; that the community accepts us as we are).

Without a 'sense of belonging', one can feel sad. It can make us feel that we are in the wrong place, that we are not supposed to be here and that we are not a member of community. This may happen e.g. if people bully us, if people ignore us or we don't agree with how the community is run. If we don't feel that we 'belong' in the community, it is possible we feel 'alienated' by the community instead.

I hope this helps, let me know if anything is unclear.