gametes, which are sex cells, carry the alleles. Why must a gamete carry one allele (represented by a letter) from each gene? meaning, why can't a gamete carry an "hh" instead of an "hs" or an "Hh" instead of a "HS?''

This is because a gamete will go fuse with another gamete and the sets of alleles from both will combine to form the required diploidy required for growth to a mature organism. If the gametes were diploid, then when two gametes fuse, the resulting zygote would be tetraploidy. Unless this is the natural ploidy for the organism - like in some plants - then the resulting zygote would not develop or the resulting mature organism would have abnormalities.