Selvage Stitches
This pattern begins with five rows of garter stitch using color B. After these rows color B is used as selvage stitches at the beginning and end of every row, framing colorwork motifs using MC and color A. Color B is not carried along the WS of the work behind the chart stitches. Instead, two separate balls of color B are used—one for each selvage edge. If you have only one call of color B you will need to wind off a relatively small amount for the other selvage edge before casting on. Be sure to twist the colors on the WS of the work when transitioning from the chart to the selvage stitches to prevent gaps.

Stranded/Fair Isle Knitting
The technique used for this square is stranded/fair isle worked flat. Colors are used in alternately according to the chart. The color not in use is carried along the wrong side of the work while not in use (this is called a float) and can be locked into place for a tidy appearance. The chart is read right to left and all stitches are knit while working the right side. The chart is read left to right and all stitches are purled while working the wrong side.

There are a number of resources on the internet where you can learn stranded/fair isle knitting. One of my favorites is this video from www.knittinghelp.com:

STRANDED FAIR ISLE KNITTING VIDEO

I like this video because she is working flat, just as you will on your square. Please watch the video all the through. In the second half she explains how to keep your floats loose to prevent puckering. If you are still having trouble with this technique I recommend that you take your project to your local yarn store for some in-person help.

Locking/catching floats
For a neat reverse side will want to periodically catch or lock the floats created when a color is not used for a stitch. For this block, the designer recommends locking the floats every other stitch. The two most common ways to catch floats are weaving and twisting. You can use the method of your choice, just be consistent throughout your work and avoid adding tension to the floats as this can result in puckering. These techniques are outlined here:

LOCKING/CATCHING FLOATS TECHIQUE VIDEO

There are many videos on the internet demonstrating these techniques. Here are a few links that demonstrate techniques used to catch floats:
Catching floats on the right side carrying yarn in both right and left hands:
CATCHING FLOATS ON THE RIGHT SIDE VIDEO

Catching floats on the right side carrying both colors in the right hand (English):
CATCHING FLOATS ON RIGHT SIDE (ENGLISH VERSION VIDEO

Catching floats on the right side carrying both colors in the left hand (Continental):
CATCH FLOATS ON THE RIGHT SIDE (CONTINENTAL)

Catching floats on the wrong side carrying yarn in both right and left hands:
CATCHING FLOATS ON THE WRONG SIDE VIDEO

If you are still having trouble with this technique I recommend that you take your project to your local yarn store for some in-person help.