Why animals and plants are separate branches, not one inside the other
It is easy to read the phrase "animal–plant" and imagine a family tree where one sits above the other. The evidence points the other way: vertebrate animals and land plants are described as evolving through somewhat different processes, shaped by different controlling factors [2]. That difference makes sense if the two groups have been on separate evolutionary paths for a very long time, each adapting to its own way of life, rather than if one had descended from the other.
A useful comparison is two distant cousins who share a grandparent but grew up in different countries. They resemble each other in family traits, yet neither is the ancestor of the other. In the same way, the shared features of animals and plants trace back to a common starting point, while the differences reflect separate histories [2].
Living side by side still changes both groups
Separate branches does not mean independent lives. Plants and the animals that eat them are locked into an ongoing back-and-forth, often called an arms race: plants evolve chemical defenses, and animals evolve ways around them [1]. One well-studied example involves a family of enzymes, UDP-glycosyltransferases, found in humans, insects, and plants. These enzymes help organisms attach sugar molecules to other substances, and the paper describes the animal–plant struggle as a major force driving how this enzyme family evolved [1].
This is co-evolution, not descent. Each side responds to pressure from the other, so the two lineages influence each other's evolution while remaining on their own branches [1]. The relationship is closer to two rivals who keep raising their game against each other than to a parent and child.
What comparing the two groups can and cannot tell us
Comparing plant and animal evolution is useful for understanding why the two kinds of life differ, but it is not a claim that one gave rise to the other. The comparison paper frames the question as one of different processes and controlling factors, not of ancestry [2]. So the comparison explains divergence, not descent.
The arms-race work likewise describes interaction and mutual pressure between plants and their animal consumers, not a line of descent from plants to animals [1]. Read together, the two lines of evidence support the same picture: separate lineages that share an origin and then shape each other through ecological contact.
So where do animals come from?
Animals come from an earlier common ancestor shared with plants, not from plants themselves. After that split, animals followed one evolutionary route and plants another, with different mechanisms and pressures at work [2]. What the two groups do share is a long history of living together and adapting to one another, which is why their stories are so often told side by side [1].
Sources used in this answer
The UDP-glycosyltransferase (UGT) superfamily expressed in humans, insects and plants: Animal⿿ plant arms-race and co-evolution
Plants and the animals that feed on them are engaged in an ongoing co-evolutionary arms race that appears to be a major force driving the evolution of UDP-glycosyltransferases and other enzymes.
Plant evolution dances to a different beat: plant and animal evolutionary mechanisms compared
Vertebrate animal evolution and land plant evolution are somewhat different processes subject to different controlling factors, consistent with the two groups being separate evolutionary branches.