It's mostly present for LINQ, when you may use an anonymous type as the projection:
var query = from person in employees
where person.Salary > 10000m
select new { FullName=person.Name, person.Department };
Here the type of query
can't be declared explicitly, because the anonymous type has no name. (In real world cases the anonymous type often includes values from multiple objects, so there's no one named class which contains all the properties.)
It's also practically useful when you're initializing a variable using a potentially long type name (usually due to generics) and just calling a constructor - it increases the information density (reduces redundancy). There's the same amount of information in these two lines:
List<Func<string, int>> functions = new List<Func<string, int>>();
var functions = new List<Function<string, int>>();
but the second one expresses it in a more compact way.
Of course this can be abused, e.g.
var nonObviousType = 999999999;
but when it's obvious what the type's variable is, I believe it can significantly increase readability.