image
The image of a transformation is the set of every output it can actually reach. Feed it all possible inputs, collect all the results, and that pile of reachable vectors is the image, also called the range.
For a matrix, the image is exactly its column space, because A*x is just a weighted blend of the columns of A. So the outputs you can produce are precisely the combinations of those columns, nothing more and nothing less.
The dimension of the image is the rank: it counts how many genuinely independent directions survive the trip through the map. A high rank means the transformation fills up a lot of the target space; a low rank means it squeezes everything into a thin slice.
Every output lands on the x-axis, so that line is the whole image; its dimension (the rank) is 1.
Kernel and image are two sides of one coin. The rank-nullity theorem ties them together: kernel size plus image size equals the number of inputs.