Why does DNA clump together?
When molecules are insoluble which means not able to be dissolved then they clump together and becomes visible.
Each cell of the body contains length of 6ft. and 2nm wide DNA. To fit all this large DNA into tiny cells, it has to be packed efficiently which means DNA twists tightly and clumps together. This makes DNA to occupy less space when packed or clumped together than it is relaxed.
When DNA is extracted from cells, it still clumps together but not that much as it is inside the cell. Therefore, when DNA is released from a cell it normally break into small fragments. When DNA is released from the cell then it is soluble in water as the phosphate group of DNA carries a negative charge that dissolves in the water.
The positive charge of sodium ions from the salt extraction solution are attracted towards the negatively charged phosphate groups on the DNA backbone. It neutralizes the DNA electric charge that allows DNA molecules to clump or to aggregate together.
When the alcohol is added then also DNA clumps together and precipitates at alcohol interface as the DNA is insoluble in alcohol that makes DNA strands to clump together.