Here's why:
* Adjectives modify nouns. In this case, "1860" modifies the noun "immigrants" by specifying *when* most immigrants were German or Irish.
* Dates can be used to describe a time period. This makes them similar to adjectives in their function.
You could rephrase the sentence to further illustrate this point:
* "During the year 1860 most immigrants were German or Irish."
In this version, "during the year 1860" acts as an adverbial phrase that modifies the verb "were." However, "1860" itself is still functioning as a part of that phrase that describes the time period.