Imagine flipping a fair coin 256 times. The particular outcome ('HTTTTHHTTTTTTTTHTHHHTHHTHTHHHHH...') is extremely difficult to replicate, but getting any outcome is very easy: just flip the coins again. In this case we also have a lot of selection bias: all the paths through DNA space that don't result in intelligent life don't result in anyone having this conversation.
Regarding the curse of dimensionality: it's a statement about the available data rapidly becoming sparse in high dimensional spaces. It doesn't really say that high dimensional spaces are necessarily sparse, it's just hard to "fill" them in with the amount of data available.
Comparing a mule with it’s parents shows how much novelty can be produced in a single generation (in this case an evolutionary dead-end of course)
Evolution implies a relatively smooth path through "DNA space" from, say for example, an early single cell eukaryote to a mushroom. However the search space is enormous. Even if we account for billions of years of evolution and a trillions of evolutionary experiments each year, a simple random walk with selection through DNA space should go nowhere because of the numbers involved. The curse of dimensionality[0] means there has to be some other principle of nature to make the search space yield a path from one viable life form to another. The search space of life would have to be 'smooth' in some sense. That 'smoothness' is something we don't understand.
If DNA space is just 256 bits (as a dramatic simplification), then 2^256 is a very very big space to search just by chance [1]. Now imagine a space orders of magnitude bigger.
[0] https://en.wikipedia.org/wiki/Curse_of_dimensionality
[1] https://youtu.be/S9JGmA5_unY?t=22 (3Blue1Browns wonderful illustration of how large 2^256 is)