Some of the ideas:
Generically (which to a mathematician means something like "most of the time"), any two planes will intersect at a line. In fact two planes either intersect at a line or are parallel and we know that two rows have to be the same (in the non-augmented matrix) for the two planes described by those rows to be parallel.
The rank $r$ is the number of planes that are linearly independent (in a sense). You have to try imagine adding planes together to get another plane in a way very similar to how you add vectors.
Then only if $r=3$ will you have a unique solution. This is the case where all three planes are linearly independent (again, in a sense).
This also should help you understand that if $r=1$ then all $3$ planes will be parallel.
If $r'>r$ then there will be no solution. I can't think of a geometric explanation (at least one in terms of planes) for this, but algebraically its because the target vector $b$ is outside the image of the transformation $x\mapsto Ax$.
Also, algebraically, $r'$ can't be more than $1$ higher than $r$ because you're only adding one more column. That column might be linearly independent of the other columns ($r'=r+1$) or linearly dependent on them ($r'=r$) but those are the only choices.