Both phenol and chloroform denature proteins making it easier to separate the protein from the soluble dna. Chloroform is also very dense so the mixture is very easy to spin down. The chloroform also stabilises the phenol and stops it partially remaining in the aqueous layer .. On centrifugation the denatured protein forms a layer on top of the stabilised phenol/chloroform layer making it easier to pipette off the purer dna in aqueous solution.
It helps that the mixture is less soluble in water so separates completely while many organic solvents have a partially ionic nature and will mix with the water and may then need further purification to remove the organic solvents but there will be many organic solvents that could be used but most people like to use tried and tested methodologies