Let us look at the addition of two vectors (red and blue in the picture) in R2. The green vector is constructed from the parallelogram.
The parallelogram tells us that the two right triangles (in gray shade) are identical. In particular, the two pink segments of the x-axis have the same length. This implies that the x-coordinate of the green vector is x1 + x2. By similar argument, the y-coordinate of the green vector is y1 + y2.