Dot product itself has no meaning. It is some sum, that we encounter in our work. Since it is long to write each time and makes equations less readable we define new expression that is substitute for long sum. Since this sum has speacial relation with respect to two vectors given ($x_1x_2+...$) we simply write in a way that makes it easier to remember.
In applications it may get special meaning when working with force etc. But again dot product itself is a just new way to denote that sum.
You may want to see where equivalent of dot product formula with norms and cosine come from. To see this form a triangle, where all sides is made up from vectors. Then use cosine formula to calculate length of one side. After simplifications, you will get that formula. Generally this where we define first time dot product, while learning this concept.