As solmax said, Blender can only handle tris and quads, nothing more than that (called ngons in the 'biz)
When you cut a quad right through, it’s easy to make 2 quads:
But if you only cut one side the face now has 5 sides:
There’s two ways to get around this. One is to leave the quad as it is, and ignore the extra vertex (that’s how I made the screenshot above) but if you move any verts you see the problem:
This hole can cause many problems - you can’t move the verts, and you get seams and/or gaps with other tools. And you can’t just patch this hole with a triangle, because then when the vert is where it belongs you have a zero-width face, which can also cause problems.
So the other alternative is that every polygon can be turned into triangles, so do that to our pentagon:
Notice that you can still move the verts around the mesh is still whole:
I hope this answers your question.
And in answer to your next question of “why doesn’t blender support ngons” the answer is that the tri/quad is too ingrained in blender’s source to adding n-gons would require a massive rewrite of large quantities of code, and no-one could be bothered.