First, let's add some labels to our axes! Suppose we have a soil sample that consists of three soil types: sand, silt, and clay.

These three together form 100% of our soil sample.

There are three axes, one for each component. In each "corner" of the triangle we're at 100% of one of the soil types. On the axis opposite to the corner, the soil type is at 0% — ie there's none of it in the sample.

On this line, soil type silt is 0%. Then our soil sample consists only of clay and sand. Likewise, on this line clay is 0%. And on this line sand is 0%

Now suppose a soil sample consists of 20% sand and 80% silt, and consequently; 0% clay., because it has to sum to 100%. The point must be located on the axis opposite to clay, because there is no clay in this sample.

This is the 80% silt line and this is the 20% sand line. The point is at the intersection of these lines.

Let's do one more, this time with all three soil types! This sample consists of 30% clay, 40% silt and 30% sand.

This is the 30% clay line, this is the 40% silt line, and this here is the 30% sand line.

Once again the point follows from the intersection of the lines. Because the three proportions cannot vary independently, really only two lines are necessary to find the point.