Start by wedging the end of your yarn into a slit with the tail toward the backside. Flip over the circle and stretch the yarn across, sliding it into the opposite notch. Flip the circle again, and this time take the yarn through the slit right next to it. Flip the circle and stretch across again, forming an X with the first string. Continue this pattern until you have what looks like bicycle spokes on the front, and little "stitches" around the edge of the backside. You'll end up with your yarn trailing down the front from the last slit.
Just start weaving in the yarn end, going around the circle with an over-under pattern.
To change colors, simply cut your yarn and begin weaving with the new color, overlapping it with the end of the old yarn so as to avoid any holes. Wrap your yarn around your hand to create a hank that you can easily pass over and under the weft.
Continue weaving until you reach your desired size.
Flip over the circle and cut through the middle of the "stitches" on the back. Take the tails that result and tie them together in a double knot, not too tight and not too loose.
To finish, just trim or weave in or otherwise secure all your ends. I like to use superglue to hold everything down on the back of the coaster. And you're done!