I was having a really hard time understanding homogenous coordinates, but watching these short videos (they are both less than 5 mins) really helped me understand. Hope this helps! :)

Links:

1. https://www.youtube.com/watch?v=JSLG8n_IY9s

2. https://www.youtube.com/watch?v=PvEl63t-opM

One way to think about homogenous coordinates (might not be particularly useful but a fun mental gymnastics exercise) is to think of our 2D image as a slice of a 3D block along a plane parallel to the xy-plane (where the 3D block is made up of a bunch of copies of our 2D image stacked on top of each other). If we want to translate our 2D image, we can think of this as shearing our 3D block (or like tilting our 3D block in the z-direction so that it looks like the Leaning Tower of Pisa) so that slices of our 3D block at different heights above the xy-plane correspond to translating by different amounts. This way, we can think of 2D translations (which are not linear) as 3D shears (which are linear, since our 3D block is still fixed at $(0, 0, 0)$). This is why the 1st and 2nd columns of our translation matrix are still $(1, 0, 0)^T$ and $(0, 1, 0)^T$, because shearing our 3D block doesn't change the slice at $z = 0$. However, the last column of the translation matrix is $(t_x, t_y, 1)^T$ because the point on our block that corresponds to $(0, 0, 1)^T$ has now been shifted to $(t_x, t_y, 1)^T$. Hope this made sense! :)