The pictures indicate how the snowflake curve evolves. Begin with an equilateral triangle. The next curve is obtained by removing the middle third of each side of the triangle and then bridging the gap with two sides of a new, smaller equilateral triangle.
Schematically, each side is undergoing a "replacement".
/\
/ \
-----> / \
________________________ ________/ \________
Similarly, once we have reached the nth iteration of the snowflake curve, the (n+1)st iteration is obtained by removing the middle third of each line segment and bridging the gap with two sides of a smaller equilateral triangle. Another way to say this is that each side undergoes the replacement pictured above, on the appropriate scale.
The snowflake is the limiting figure obtained by this procedure.
You can see that the area is finite by drawing a circle around the figure. For a nice exercise, see if you can write a geometric series for the area and find the sum.
To see that the length of the boundary is infinite, just notice that the length of the curve at iteration n is 4/3 times the length of the curve at the previous iteration, as is apparent from the "replacement" sketch.