Why does the PKCS1 RSA private key structure contain more than just exponent and modulus?
It has to do with optimizing RSA. 

It turns out that using the Chinese Remainder Theorem with $p$, $q$, $d\pmod{p-1}$, and $d\pmod{q-1}$ (i.e., prime1, prime2, exponent1, exponent2 from the data structure in the question) to run the decryption operation faster than if you only had $d,n$.

For more information on how it is done, I found this reference http://www.di-mgt.com.au/crt_rsa.html