How do I pass an extra parameter to the callback function in Javascript .filter() method?

Make startsWith accept the word to compare against and return a function which will then be used as filter/callback function:

function startsWith(wordToCompare) {
    return function(element) {
        return element.indexOf(wordToCompare) === 0;
    }
}

addressBook.filter(startsWith(wordToCompare));

Another option would be to use Function.prototype.bind ^[MDN] (only available in browser supporting ECMAScript 5, follow a link for a shim for older browsers) and "fix" the first argument:

function startsWith(wordToCompare, element) {
    return element.indexOf(wordToCompare) === 0;
}

addressBook.filter(startsWith.bind(this, wordToCompare));

---

I dont really understand how the default parameters it takes are passed

There is nothing special about it. At some point, filter just calls the callback and passes the current element of the array. So it's a function calling another function, in this case the callback you pass as argument.

Here is an example of a similar function:

function filter(array, callback) {
    var result = [];
    for(var i = 0, l = array.length; i < l; i++) {
        if(callback(array[i])) {  // here callback is called with the current element
            result.push(array[i]);
        }
    }
    return result;
}

The second parameter of filter will set this inside of the callback.

arr.filter(callback[, thisArg])

So you could do something like:

function startsWith(element) {
    return element.indexOf(this) === 0;
}
addressBook.filter(startsWith, wordToCompare);

For those looking for an ES6 alternative using arrow functions, you can do the following.

let startsWith = wordToCompare => (element, index, array) => {
  return element.indexOf(wordToCompare) === 0;
}

// where word would be your argument
let result = addressBook.filter(startsWith("word"));

Updated version using includes:

const startsWith = wordToCompare => (element, index, array) => {
  return element.includes(wordToCompare);
}