For complex multicelluar organisms it's not possible to demonstrate (yet). The best you can hope for is an extensive logical chronosequence of many fossils.

One set of fossils makes us reasonably suspicious. that what we see is relatedness, not change over time. So we notice that some fossil sequences can be found in a series of geological beds. Each bed can be atomically aged by knowing the half-lives of various isotopes and their decomposition products. So for the fossil sequence to be arranged in a chronosequenced geological formation we have to suppose that the creature was able to burrow into the various formations without leaving indications of their burrows or we must assume the fossils were formed in situ, in a time-sequence.

Now multiply that by many, many thousands of different lineages of different plants and animals across numerous phyla. Patterns emerge. Paleontologists look for causes of patterns formed by multiple lineages. They use nuclear physics, chemistry, and geology to infer climate variation, volcanic episodes, and atmospheric changes taht correlate with patterns in fossil lineages.

What almost uniformly happens when there is a great burst of diversity and multiple new lineages or a sudden decimation or extinction of many lineages is some environmental factor. Thus, environment precipitates changes in the fossil lineages.

This is evolution through natural selection over geologic time. It isn't iron-clad causation. But it is the most parsimonious explanation that does not require a miracle. Since miracles are the perview of God and Religion, not science, scientists evaluate scientifically tractable problems.