
Before defining the surface area, I want to start from the bottom of this concept. Kids start to learn about points in grade five or six. Points can be compared to real numbers. For example, the dot we use to mark the period (full stop ".") is a point. A point can be represented by a capital letter. Following are two points A and B in space: A . . B When we put trillions of points side by side we get a line segment. Hence points give birth to a line. Now if we join the above two points we get a line AB. Hence, a line segment can be represented using two capital letters.
There are many examples of lines in daily life, such as a sewing thread, an electric wire can be compared to a line. As billions of points construct a line, similarly many lines on a piece of paper or on land can give rise to the concept of area. Lines has only one way to go at a time, which means a line can only be top to bottom (vertical) or left to right (horizontal) but it can't be both. Hence lines have only one dimension and we call it "the length". As we measure other quantities, similarly, we can define units to measure the length of a line. We can measure the length of lines in metres, centimeters, millimeters, inches, yards or even in miles or kilometers. Below are some examples of different lines: Look at above lines, line segments, right angles, rays and also a spider web containing a bunch of lines and curves. 

Hope it has cleared the idea of dimensions in kids' minds. If kids get the idea of dimensions, and know that lines are only one dimensional then they are ready to learn area and then surface area. Now, if we draw two parallel lines vertically and two horizontally and let them intersect we get a very simple geometric shape called a rectangle as shown below: So, four lines when cut each other in a specific order, they make a rectangle. Look at above rectangle, it can be measured two ways, left to right (horizontal line called length) and top to bottom (vertical line called breadth or width). Therefore a rectangle have two dimensions called length and breadth (width). A page of a note book or textbook is the simplest example of a rectangle. Below are some more example of two dimensional shapes: 
Hence the lines give birth to a shape. Remember we are talking about straight lines. A curve is also a line but not straight. Kids can draw more shapes using lines. Here I want to stop; when lines make a two dimensional shape, they occupy some space on the surface over where they have drawn. This space bounded between lines is called "the area" or "surface area" in case of solid three dimensional shapes. 
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