Some population genetic inference methods require the data to be phased because they look for long stretches of matching alleles between individuals, rather than just matching genotypes. Here, 'genotype' data is un-phased, and 'allele' data is phased.
This is because there is a degree of ambiguity when it comes to matching heterozygous positions - if the data is un-phased, you don't know if 2 individuals share mutations on the same chromosome (here, I mean different copies of the same chromosome - like either copy of chromosome 1, for example). This is important, because the fact that the mutation is shared on the same chromosome means that they are much more likely to share recent common ancestry with one another.
Let's say we have genotype data for two individuals (where the number corresponds to the number of non-reference alleles present):
Ind 1
0 ----- 1 ----- 2 ----- 0
Ind 2
0 ----- 1 ----- 2 ----- 0
If were looking at unphased genotype data only, we know the alleles match exactly in the 1st, 3rd and 4th positions, because they are homozgyous (either 0 or 2 copies of the non-referencee allele), but because the second position is heterozygous, we can't be sure which chromosome the non-reference allele is on. This is because the haplotype configurations for individual 1 could be either
0 ----- 0 ----- 1 ----- 0 Chromosome1a
0 ----- 1 ----- 1 ----- 0 Chromosome1b
or
0 ----- 1 ----- 1 ----- 0 Chromosome1a
0 ----- 0 ----- 1 ----- 0 Chromosome1b
i.e., we don't know which chromosome the non-reference allele at position 2 is on.
If we then phased the data, and found that the non-reference allele was on chromosome 1a for Individual 1, and chromosome 1b for Individual 2 there is less evidence of recent shared ancestry than otherwise. I think this is more or less the approach that 23andme takes when they are inferring your relationship to other individuals in the dataset. After writing this, I realised there is a much better description of what I just said on the 23andme website!
For example, the above applies to the ChromoPainter/fineSTRUCTURE algorithms. This means that they are able to detect more subtle genetic differences between individuals than if you were to use non-phased data.
This is a slightly niche example, and there occasions where you need phased data, but this is the one I am most familiar with :). Off the top of my head, methods such as Multiply Sequential Markovian Coalescent (MSMC), RELATE, and Extended Haplotype Homozygosity all require the data to be phased.