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I know the traditional method for finding an image of a matrix is finding the pivot columns of the rref of the matrix and then the corresponding columns are the basis of the image, but I'm wondering if this other method works. You take the transpose of a matrix, find the rref of the transpose, and then the rows become the basis for the image. For reference, here's an instance where it's used: Image and kernel of a matrix transformation

There wasn't an explanation given in that post so could someone help me come up with a formal proof for this method? If this method doesn't always work, could someone give an explanation or counter example?

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Yes, this always works. If you column-reduce a matrix, the non-zero columns that remain form a basis for the image. The reason is that when you column-reduce a matrix $M$, you end up with a matrix $C=MT$, where $T$ is an invertible matrix. Then $\operatorname{im}C=\operatorname{im}M$ and by construction the $\operatorname{rank}M$ non-zero columns of $C$ are linearly independent.

You might also be interested in having a look at this question, in which bases for both the image and kernel are found simultaneously by column reduction.

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