Because that's how hot the Sun (or more precisely, its photosphere) is. Our eyes are evolved to see things in the peak intensity range of sunlight, which hits specifically on the Balmer series.

A star in the Main Sequence is going to have a temperature roughly in the range where most of the black-body radiation is in the 1-3 electron volt range of photon energy. Well, the Lyman series going down to the $n=1$ level involves transitions of 10 eV or more given hydrogen's total ground-state ionization energy of 13.6 eV, and the Paschen series going down to the $n=3$ level is basically all under $1eV. Like the famous porridge in the children's tale, the Balmer series is just right for the 1-3 eV range where most of radiation, and the solar radiation we are evolved to see, is emitted.

Because that's how hot the Sun (or more precisely, its photosphere) is. Our eyes are evolved to see things in the peak intensity range of sunlight, which hits specifically on the Balmer series.

A star as massive as our Sun in the Main Sequence is going to have a temperature roughly in the range where most of the black-body radiation is in the 1-3 electron volt range of photon energy. Well, the Lyman series going down to the $n=1$ level involves transitions of 10 eV or more given hydrogen's total ground-state ionization energy of 13.6 eV, and the Paschen series going down to the $n=3$ level is basically all under 1 eV. Like the famous porridge in the children's tale, the Balmer series is just right for the 1-3 eV range where most of radiation, and the solar radiation we are evolved to see, is emitted.