A young black American soldier leaves his home and family to cross the
ocean to put an end to the German nightmare.
Using home and family in this sentence adds extra weight to the soldier's sacrifice/decision. It is stating he is leaving both a place he is familiar with (his home) as well as the people he cares about/people who care about him (his family). It highlights that he is uprooting himself both physically as well as socially.
In regards home vs family:
Home and family can have emotional connotations:
Home = A place that you live / have lived (probably long-term) that you have an emotional connection to.
Family = (Mostly) permanent group of people (usually blood-related or by marriage) with whom you have a emotional connection.
Home can mean simply the place that you live. It can also mean a special place that you live/ have lived. Many people feel at home, regardless of the actual location, when they are with their family. Thus, they may consider, their 'true' home, their family.
For example, a person who moves to a different city for a new job would live permanently in that new city. The house in that city that they live in would be their new 'home'. However, if that person felt a strong tie to their parent's house, for example, they may still consider their parent's house their 'home' despite no longer actually living there.
Less emotionally-charged phrasing for 'home' could be: 'his residence/house/apartment' and as an alternative for family: 'his relatives'.