Why does:
public class Addition { 
  public static void main() { 
    int a = 0; 
    double b = 1.0; 
    a = a + b;
    System.out.println(a); 
  }
}
        double + int returns double, so double = double + int is legitimate, see JLS 5.1.2 Widening Primitive Conversion on the other hand int = double + int is "Narrowing Primitive Conversion" and requires explicit cast
int = int + double is essentially
int = double + double
and you cannot do that without casting...
The int += double forces the result to an int while the other one requires casting.
So a = (int)(a + b);
should compile.
Edit: as requested in the comments... here is a link to more reading (not the easiest read, but the most correct info): http://docs.oracle.com/javase/specs/jls/se7/html/jls-15.html#jls-15.26.2
In Java += operator has an implicit cast to the left hand type. This goes for all composed operators.
As everyone already stated, the += has an implicit cast. To help illustrate that, I'm going to use an app I wrote a while back that is perfect for these types of questions. It's an online disassembler so you can check out the actual bytecode that's being produced: http://javabytes.herokuapp.com/
And a table of their meanings: http://en.wikipedia.org/wiki/Java_bytecode_instruction_listings
So let's take a look at the bytecode from some simple Java code:
int i = 5;
long j = 8;
i += j;
Disassembled code. My comments will have a // in front.
   Code:
        0: iconst_5  //load int 5 onto stack
        1: istore_0  //store int value into variable 0 (we called it i)
        2: ldc2_w #2; //long 8l
                     //load long 8 value onto stack.  Note the long 8l above
                     //is not my comment but how the disassembled code displays 
                     //the value long 8 being used with the ldc2_w instruction
        5: lstore_1  //store long value into variable 1 (we called it j)
        6: iload_0   //load int value from variable 0
        7: i2l       //convert int into a long.  At this point we have 5 long
        8: lload_1   //load value from variable 1
        9: ladd      //add the two values together.  We are adding two longs
                     //so it's no problem
        10: l2i      //THIS IS THE MAGIC.  This converts the sum back to an int
       11: istore_0  //store in variable 0 (we called it i)