For a heterozygous diploid individual with genotype $Aa$, the two alleles normally lie at corresponding loci on the two homologous chromosomes. During meiosis I, those homologs segregate to opposite daughter cells.

As a result, each haploid gamete receives only one allele copy at that locus:

Aa diploid parent
      ↓ meiosis
A gametes or a gametes

If homolog segregation is unbiased, the two gamete classes are produced with probabilities

$$P(A)=\frac12,\qquad P(a)=\frac12.$$

This is the chromosomal basis of Mendel's law of segregation: the two allele copies carried by a diploid individual separate during gamete formation so that a gamete receives one or the other, not both.

Fertilization restores allele pairs

A gamete from one parent then combines with a gamete from another. For an $Aa\times Aa$ cross, each parent contributes either $A$ or $a$.

The four equally likely gamete combinations are

A × A → AA
A × a → Aa
a × A → Aa
a × a → aa

so the genotype probabilities are

$$P(AA)=\frac14,\qquad P(Aa)=\frac12,\qquad P(aa)=\frac14.$$

The familiar $1:2:1$ genotype ratio is therefore not an unexplained numerical pattern. It follows from equal homolog segregation in each parent plus random pairing of gametes at fertilization.

Segregation applies to alleles, not phenotypes

Meiosis separates chromosome copies carrying alleles. It does not directly sort visible traits. A later genotype-to-phenotype relationship determines whether two different genotypes produce distinguishable phenotypes.

The mechanistic chain is

homologous chromosomes separate
        ↓
alleles segregate into gametes
        ↓
fertilization combines gametes
        ↓
offspring genotypes
        ↓
phenotype depends on how those genotypes function

Mendelian segregation is therefore a direct statistical consequence of chromosome behavior during meiosis; it does not depend on any particular relationship between genotype and phenotype.