Step 1: Insert the needle, from back to front, up through one hole. Leave a small tail on the wrong side of your sweater.
Step 2: Push the needle through the other hole of the button, down through the knitting to the back of the sweater. Pull tight.
Repeat these two steps 2 to 3 more times.
Step 3: Flip your sweater over so you can work on the back. You'll notice a tight loop has resulted from Steps 1 and 2.
Step 4: Insert your needle through the loop we saw in Step 3. From below, insert it through and out the other side. Repeat this step twice more.
Step 5: This is my own little extra step. Cut your thread so that it is the same length as the tail. Place both tails into the eye of your needle together. Weave those two ends through one or two stitches of your knitting, just to secure them.
Step 6: Snip the tails a centimeter or two away from the knitting -- that will allow the ends to "settle" into the knit.