How does gene editing in one cell affect other cells?
Gene editing is not done on a single cell but a bunch of cells. However, it still does not affect all the cells. Gene editing/therapy for the entire organism works in these cases:


Stem cells are edited, which give rise to many other somatic cells. Usually done in case of gene therapy in which the stem cells are obtained from the patient, edited and re-implanted. 
Some embryonic cells are edited and reimplanted in the developing embryo which gives rise to what is known as a chimeric organism (some cells in the body carry the edit while others don't). Now some of the edited cells will give rise to germ cells which in turn give rise to gametes. In the next round of breeding the chimeric parents would give rise to a fully transgenic (with edited gene) organism.
For unicellular organisms, it is pretty obvious: you just use a selection marker or extensive screening to select for the cells carrying the edited gene.