A Line Integral Is kind of like a fence... let me try my best to explain.
with a regular integral of one variable you are simply finding the area under the curve or function you are trying to integrate, however now that you are dealing with line integrals you have more than one variable.
so what i mean by a fence, picture a surface floating above the xy plane, if you were standing on the xy plane and looked up you would see a curvy (or even flat) ceiling which is biased on your given functions. Now you are trying to use a line integral to calculate the area of a wall or fence of sorts with its base on the xy plane that is as tall as the distance from the plane to the surface above, however not all fences are perfectly straight lines, instead the fence follows a function or curve on the xy plane.
so you are summing the infinitely small areas of a fence. Which consists of a curve drawn on the xy plane and then extruded upwards into a until it hits the surface above the xy plane which is given by a function of another variable.
Hope this long winded explanation is of some help, its kind of hard for me to explain in words, just try to draw a picture of sorts, it is much the same as integrals of one variable accept now the bottom of your area is not constrained to an axis but is free to follow a curve around a plane (or even another surface)