There are 2 ways to do a Moebius.
One is to use a ling circular knitting needle and a special cast on. With this method, you knit it in the round and the half twist is created when you set up the cast on.
The other way is easy but requires putting the half twist in after knitting a straight piece and then grafting the stitches.
Let's see if I can explain this:
A____________________________________B
C...........................................D
Please ignore the dotted line, I had to do that to make the C and D show up in the approximately correct places.
The line represents a long rectangular piece of knitting.
A,B,C, and D are the four corners of the rectangle.
Cut a long rectangular strip of paper, mark the four corners the same way on both sides of the strip (a back to back with a, etc) as shown and follow along
With the A C side to the left, fold the left end in about halfway towards the center. Then pick up the right b-d end like you're going to fold it in BUT give it a half of a twist so that the upside down D meets the right side up A and the upside down B meets the right side up C. Tape the paper at the join.
Now, Run your fingers around the edge of the paper, starting at the upside down D. You'll see that you can run your fingers all around the edge and come back to the same point without taking your fingers off the paper. The strip has one edge. That is a Moebius strip
Putting that twist into your flat knitted scarf before grafting the first row to the last row will make a Moebius scarf.