How is StringBuffer implementing append function without creating 2 objects?

First there is a problem with your question:

String s = "orange";
s.append("apple");

here two objects are created

Correct, two Objects are created, the String "orange" and the String "apple", inside the StringBuffer/StringBuilder no Objects will be created if we don't overflow the buffer. So those lines of code create 2 or 3 objects.

StringBuilder s = new StringBuilder("Orange");
s.append("apple");

Now here only one object is created

I don't know where you get that, here you create one StringBuilder Object, one "Orange" String, one "apple" String, for a total of 3 Objects, or 4 if we overflow the StringBuilder buffer. (I count the array creation as object creation).

I read your question as, how can StringBuilder do the append without creating a new Object (when the buffer is not overflown)?

You should look at StringBuilder, since it's the non thread safe implementation. The code is interesting and easy to read. I've added the inline comments.

As internal structure there is a char array, not a String. It is initially built with length 16 and will be increased every time the capacity is exceeded. If the Strings to append fit within the char array, there is no need to create new Objects.

StringBuilder extends AbstractStringBuilder, where you'll find the following code:

/**
 * The value is used for character storage.
 */
char value[];

Since not all the array will be used at a given time, another important variable is the length:

/**  
 * The count is the number of characters used.
 */
int count;

There are many overloading of append, but the most interesting one is the following:

public AbstractStringBuilder append(String str) {
    if (str == null) str = "null"; //will literally append "null" in case of null
    int len = str.length(); //get the string length
    if (len == 0) return this; //if it's zero, I'm done
    int newCount = count + len; //tentative new length
    if (newCount > value.length) //would the new length fit?
        expandCapacity(newCount); //oops, no, resize my array
    str.getChars(0, len, value, count); //now it will fit, copy the chars 
    count = newCount; //update the count
    return this; //return a reference to myself to allow chaining
}

String.getChars(int srcBegin, int srcEnd, char[] dst, int dstBegin) Copies characters from this string into the destination character array.

So, the append method is quite simple, the only magic left to discover is the expandCapacity, here it is:

void expandCapacity(int minimumCapacity) {
    //get the current length add one and double it
    int newCapacity = (value.length + 1) * 2; 
    if (newCapacity < 0) { //if we had an integer overflow
        newCapacity = Integer.MAX_VALUE; //just use the max positive integer
    } else if (minimumCapacity > newCapacity) { //is it enough?
        //if doubling wasn't enough, use the actual length computed
        newCapacity = minimumCapacity;
    }
    //copy the old value in the new array
    value = Arrays.copyOf(value, newCapacity); 
}

Arrays.copyOf(char[] original, int newLength) Copies the specified array, truncating or padding with null characters (if necessary) so the copy has the specified length.

In our case, padding, since we're expanding the length.