It was important that Mendel examined not just the F1 generation in his breeding experiments, but the F2 generation as well because …?

Through his experiment with F2 breeding, Mendel created his 3rd Law (Law of Dominance), although rather considered as a principle since it’s not always applicable.
Through Mendel’s third law he explained the behavior of dominant and recessive alleles.
Dominant alleles are the ones that are expressed (shown), while recessive alleles are suppressed (hidden) by the dominant alleles.
For example in his mono hybrid cross experiment with peas, the dominant (AA) peas were smooth, while the recessive alleles (aa) were wrinkled. This is called the parental generation or P-Generation, that when crossed with each other give rise to the F1-Generation.
P-Generation: AA x aa
F1-Generation: Aa Aa Aa Aa
F2-Generation: AA Aa Aa aa

In the F1-Generation all of the products contain a dominant allele (A), therefore (since it’s the allele that is expressed) the peas all have the same phenotype (smooth) and the same genotype (Aa). The F1-Generation is uniform .
In the F2-Generation we can see that three of the products contain at least one dominant allele, therefore three of the four (AA, Aa, Aa) will be smooth, two will have the exact same genotype (Aa,Aa) and one of the four will contain two recessive alleles (aa), therefore since these alleles aren’t “masked” by dominant alleles, they WILL express their (wrinkled) phenotype.
The reason his experiment with the F2-Generation was so important, is because it explained why many occur in the human phenotype or genotype, for example sickle-cell disease which is an autosomal recessive disorder , meaning that the carrier of this disease is a recessive allele and is expressed when combined with another such as in the F2-Generation ( aa ), meaning that this individual will suffer from sickel-cell disease.