If you connect the 5 points of a regular star, you have a regular pentagon. This will have a bunch of isosceles triangles inside of it. We can use these relationships to solve for what you want. (Please draw this so that you can follow what I am doing.)
Each interior angle of a regular pentagon measures (5-2)*180/5 = 180*3/5 = 108 degrees.
Call the measure of the angle at the point of the star n degrees.
Call the measure of one of the angles inside the pentagon, adjacent to the point of the star x degrees.
Then 2x + n = 108.
Notice that you also have a large triangle connecting the point of the star to the opposite side of the pentagon. From that triangle, you should see that 2x + 3n = 180. Solve this system of equations.
2x + n = 108
n = 108 - 2x
2x + 3n = 180
2x + 3(108 - 2x) = 180
2x + 324 - 6x = 180
-4x = 180 - 324 = -144
x = 144/4 = 36
So each angle at the point of a regular 5-pointed star measures 36 degrees. Then the sum of the 5 would be
5*36 = 180 degrees
For a second method, you could actually make a 5-pointed star on paper (even if it is not regular). Mark each of the corners, and then tear them off. If you put them next to each other on a desk so that the points touch, without leaving any spaces between the sides of the angles, then they should form a straight line. Since you get a straight line, this is 180 degrees.
I hope this helps!