Our germ cells are haploid, while our somatic cells are diploid. The diploid cells of a child result from the fusion of a paternal haploid germ cell and a maternal haploid germ cell. When these germ cells are generated during meiosis, the paired diploid chromosomes of each parent (which they in turn received one of from each parent) are randomly segregated such that for a given chromosomal pair there is an approximately equal chance of a given germ cell containing either member of the pair independent of the what other chromosomes it receives.
In the case of a single gene with a single copy on both the maternal and paternal chromosomes, the Punnet Square accurately estimates the probably of inheriting each possible genotype. (50% chance of getting one of two alleles from dad * 50% chance of getting one of two alleles from mom). In the case of multiple genes on different chromosomes, the 50% chance of any given allele still holds, as all the genes segregate independently. The accuracy of the Punnet Square fails when genes are sufficiently close to each other on the same chromosome. Genes distant from each other on the same chromosome still segregrate independently due to a high level of recombination events between chromosomes in meiosis I, but genes in close physical proximity no longer segregate independently. The closer two genes are physically the more they tend to co-segregate. This tendency is reported empirically as a "distance" in centiMorgans and is related to physical distance in base pairs (kB or mB) but is not completely proportional to physical distance due to the existence of recombination "hot spots."
It's important to note that is accurately predicts genotypic ratios, but not necessarily phenotype. Phenotypes can be less than 100% penetrant, can cause embryonic lethal, epistasis, etc. Such that the observed phenotypic ratio does not match the predicted ratio. (For example, if YY was known to be 100% embryonic lethality, live births -> observed phenotypes instead be expected at 1/3 of each of the remaining genotypes. If Y was instead phenotypically dominant (one copy of Y determines phenotype), then 75% of children would be expected to be phenotypically Y (genotypically, the Y phenotypic children would be expected to distribute evenly between YO, YY, BY).