The seed was still alive for those 1000 years. Still respirating from stored fats and carbs. Still performing cellular activities like DNA replication and repair.

Also plants often have many duplicate copies of their genome per cell.

Here's how I understand it, though I am just a layperson;

As all the strands of DNA in the cells of the seed degrade in different ways, the cells can still patch up the damaged DNA.

I assume 1/4th of the DNA can replicate enough after 1000 years.

Most of the DNA is non coding. Damage to it almost never compromises the genome.

This neat trick works like the sacrificial metal on the hulk of ships. (Loose analogy)