Me too I get confused sometimes about this form.
In English we would think as 'We have 5 persons in our family' but what you've wrote is literally 'our family is 5 persons'. It sounds less natural but in Korean that is perfectly fine.
As Tony said 이다 has a sense of coupling, for instance 저는 프랑스인입니다 (저 = 프랑스인). As well as 우리 가족은 다섯 명입니다 (family = 5 persons).
Don't be surprised you'll see this form in other sentence, I could give you an example, for instance :
우리는 시내 가까이에 새 사무실을 열 계획이다.
We plan to open a new office near the downtown area.
If you think literally from English you would want to write '계획이 있다' (we have the plan). But in Korean '계획이다' (Is plan) is correct because as I said previously it follows the logic of 'open a new office near downtown = plan'.
Hope this adds some help for your understanding.