Explain how to use the standard normal table to find the probability associated with the shaded area under the curve.

Attached is an extract of a normal probability curve, with a table below it.

The curve itself is symmetric about a point where z=0, i.e. the "middle of the curve.  The TOTAL area under the whole curve is 1.0 exactly, from z=-infinity to z=+infinity.

The value of the table below indicates the AREA UNDER the curve and to the LEFT of the indicated value of Z.

For example, for a value of Z=1.9, the area of the curve to the left of Z=1.9 can be read as 0.9713 (accurate to 4 decimals).
Similarly, the area to the left of Z=0.4 can be read as 0.6554.

Since the required orange area (from the question) is between 0.4<=Z<=1.9, we only have to subtract the two value we read earlier, 
i.e. P(0.4<=Z<=1.9)=0.9713-0.6554=0.3159
More accurate tables will give the value 0.3158617.

The other columns are used when the values of Z is accurate to 2 places after the decimal, e.g. for Z=1.54, P(1.54)=0.9382 (highlighted in yellow in the table.)

Hope this helps, and please post in comments if you have questions.