What does the code query.Take(() => 1) do?

There's a few things to explain here.

1. query.Take(() => 1 (docs)

   This returns an IQueryable<T> which may contain a single item. It will run a query that uses the SQL Server TOP syntax which means it will work on SQL Server 2008 and earlier.

2. query.Take(1)

   This is similar to the previous version, but it uses the newer SQL Server OFFSET/FETCH syntax and is the preferred method unless you need legacy support.

3. query.FirstOrDefault()

   This will return a single instance of your entity, or a null if there are no matches. If you want a single item, this is usually the preferred method. You will need to check for null in your code though.

will return an enumerable whereas query.FirstOrDefault will give you a single item, or null. They will run very different

The code writer should have written documentation of that line.

But if you use MSFT documentation:

https://docs.microsoft.com/en-us/dotnet/api/system.data.entity.queryableextensions.take?view=entity-framework-6.2.0

public static System.Linq.IQueryable<TSource> Take<TSource> (this System.Linq.IQueryable<TSource> source, 
    System.Linq.Expressions.Expression<Func<int>> countAccessor);

countAccessor Expression<Func<Int32>>

An expression that evaluates to the number of elements to return.