Why are there transposing instruments

Music is often written in transposed form for these groups of instruments so that the fingerings correspond to the same written notes for any instrument in the family, even though the sounding pitches will differ.

1. Why are instruments pitched in different keys?
2. Why do clarinets have to transpose?
3. Why do French horn players have to transpose?

Why are instruments pitched in different keys?

This happens because the horn traditionally did not have valves so the composers wrote the parts in different keys to accommodate the music. The horn player would have to add additional pipes (or crooks) to pitch the horn differently for different pieces or different movements within the same piece.

Why do clarinets have to transpose?

Transposing instruments make it easier to switch between instruments in the same key since the musician won't have to learn new fingerings. Having transposing instruments also makes reading music in concert key much easier. Many orchestras require the clarinetists to play an A clarinet.

Why do French horn players have to transpose?

The reason behind the French horn's transposition is similar, in that it's to do with the size of the instrument. In the past, before valves, the French horn could only play "in" one key (technically, they could only play notes in the harmonic series based on the fundamental pitch of the instrument).