how are pictures digital enlarged?

The enlarging technique is called resampling. In your example, if you were to take the 960x720 output grid and lay it over the 640x480 input pixel grid, it would define the points of interpolation between the original pixels. There are many different ways to do the interpolation, but the three most common are:

Nearest neighbor (just pick the closest pixel) - fastest but poor quality

Bilinear (linearly interpolate between the 2x2 surrounding pixels) - fast, but not very sharp

4x4 interpolation with various interpolating functions (Kernels). This is known by names such as Bicubic, cubic convolution, lagrange interpolation, Lanzos, Bspline and others. This is the highest quality, but slowest technique, since it requires a 4x4 multiply-add at each output pixel. But modern processors are so fast now that it's not a problem. The processors also have special hardware to speed this up. That's one of the functions of the SSE instruction set extensions.

Upsampling (enlarging) is pretty easy. Downsampling (reduction) requires some kind of anti-alias filtering to be performed before interpolation, although in practice, the anti-alias filter is sometimes combined with the interpolation filter if the degree of reduction is less than 2X or so.

I doubt a picture from a phone has a lot of quality to begin with, but if you have photoshop or some other drawing program you can open the picture and increase the resolution (100 or higher, like 300) which usually increases the size of the picture as well.