Create an array with the highest values from each array

The challenge says that you will be given exactly four subarrays, but it doesn't specify the length of each subarray. You've assumed a square matrix, which is not stated in the problem.

To find the maximum of 4, 5, 1, and 3, you could write Math.max(4, 5, 1, 3). If those numbers are given to you as an array, you could use Math.max.apply(null, array).

To transform each element of the outer array, use arr.map().

function largestOfFour(arr) {
  return arr.map(function(subArr) {
    return Math.max.apply(null, subArr);
  });
}

console.log(largestOfFour([[4, 5, 1, 3], [13, 27, 18, 26], [32, 35, 37, 39], [1000, 1001, 857, 1]]));

Bug

  for(var i = 0; i < arr.length; i++){
    for(var k = 0; k < arr.length; k++){   
      console.log(arr[i][k]);

There is a bug here.

  for (var i = 0; i < arr.length; i++) {
    for (var k = 0; k < arr[i].length; k++) {   
      console.log(arr[i][k]);

It only worked because both dimensions were the same. But that doesn't have to be true.

I would also prefer more whitespace to make it easier to read the code.

Nitpick

      console.log(arr[i][k]);

This looks like debugging code to me. You should remove debugging code before sending out for review (or going into production).

Avoid stomping globals

      a = arr[i][k];

If there is an a variable that is already visible in this scope, this will assign over that. That won't break anything in this code, but it could break other code, as its variables may change unexpectedly.

      var a = arr[i][k];

If you use var, it will create a new variable in this scope.

      if(a>largestNum){
        largestNum = a;

In this case, even if you don't switch to the @200_success solution, it would be even easier to just get rid of a.

      if (arr[i][k] > largestNum) {
        largestNum = arr[i][k];

Don't do extra work

You can simplify your interaction with largestNum:

function largestOfFour(arr) {
  var newArray = [];
  for (var i = 0; i < arr.length; i++) {
    var largestNum = 0;
    for (var k = 0; k < arr[i].length; k++) {   
      if (arr[i][k] > largestNum) {
        largestNum = arr[i][k];
        newArray[i] = largestNum;
      }
    }
  }

  return newArray;
}

This both reduces the scope and reduces the number of assignments. You clear largestNum one more time than is actually necessary.

Naming

/* Given a 2D array, return the maximum value from each row.  */
function findMaximums(arr) {
  var maximums = [];
  for (var i = 0; i < arr.length; i++) {
    maximums[i] = 0;
    for (var j = 0; j < arr[i].length; j++) {
      if (arr[i][j] > maximums[i]) {
        maximums[i] = arr[i][j];
      }
    }
  }

  return maximums;
}

I prefer a descriptive name like maximums to a generic name like newArray.

It is more common to use j as a second loop variable and k as the third.

I also prefer findMaximums to LargestOfFour, although that may be set by the challenge.

This seems like the simplest version of your original algorithm.