what is the importance of dna in darwin's theory of natural selection?

Darwin proposed the obvious descent with modification over many generations was due to the mechanism of selection resulting from the natural limitations of resources to a population with a stable size but producing more offspring than the habitat could support. The predicted result was a differential reproductive success with the most suitable characteristics becoming more common.

This was fine as far as it went but without understanding how the characteristics passed between generations it was of limited practical use. The first useful application was in understanding the nature of native European grape vine vulnerability to the phylloxera aphid while the aphid was harmless to American grape vines adapted to the pest. However without the and population biology mathematics little more than a slight understanding of community relationships was possible.

The modern synthesis of concepts has provided a usable theory of gene pool changes in allele frequencies in populations.

Unimportant.
Darwin knew absolutely nothing about DNA.
Heck, it wasn't until 1944 that it was demonstrated that DNA was a molecule of heredity.

If your teacher asked this question in exactly this way, your teacher is really silly.