Why are punnett squares useful in genetics?
They are useful as they can predict the genetic probability of a particular phenotype arising in a couple's offspring. In other words, it can tell you if you will or will not have a certain trait.
How does this work?
Well, first off you should know that each person inherits two versions of the same chromosome - one from mom and one from dad. Hence, can receive different versions of the same genes, or different alleles.
Now what happens if you get two versions of the same allele? Well, there is always a dominant allele, and a recessive allele. The dominant allele always shuns out the recessive one, so the only way the recessive allele can be expressed is if an individual inherits two recessive alleles.
What a punnett square does is that it tells you, given the genotypes of the parents, what alleles are likely to be expressed in the offspring.
The classic example of this would be Mendel's peas. For pod color, the pea plants had two different alleles: Green and Yellow. Yellow is dominant to green. Hence, let's call the yellow allele "Y" and the green one "y".
Now if you were asked the probability of an individual with a Yy configuration and an yy individual breeding to produce a green offspring, then you'd use a punnett square.
This is how it'd look:
We know that each individual can give only one of it's two alleles. Therefore, we have to calculate what would happen in each case. And that is what a punnett square does.
If you're still confused, try watching this video by Bozeman Science.
Hope that helped :)