In the modern age? Cells beget cells, and proteins for DNA management are transferred during cell division.
If you’re asking about evolution going further back, RNA can form structures that act like enzymes (an example: ribosomes, which make proteins, have enzymatically active RNA structure). It is hypothesized that early life used RNA for information storage and for structure/enzymatic activity.
DNA is more stable than RNA, so, the logic of this hypothesis is there was eventually evolutionary selection for organisms which formed DNA as their primary information storage. Similarly, proteins can be more stable than RNA.
It should be noted that this is a pretty active area of research, so our understanding of this topic is also regularly being updated.
This is a very simplified answer since this is ELI5!
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