Why do proteins targeted for chloroplasts require two signal peptides?
Protein import to chloroplasts (and also mitochondria) can still have multiple destinations, because these structures themselves have sub-compartments:


both have an inter-membrane space as well as an 'inner space' (the stroma in chloroplasts, matrix in mitochondria).
proteins can also be targeted towards either of the two membranes
additionally chloroplasts have the Thylakoid stacks in the stroma which are separated by another membrane layer.


Due to this complex structure you need multiple signal peptides to properly target proteins to their destination.

I don't remember (or can find) all the details now but basically you (can) have:


a 'main' signal peptide that targets proteins directly towards the stroma (which will be removed there by the stromal processing peptidase (SPP)).
an additional signal peptide for import to the the thylakoid stacks (which will be 'made avaibalble' by cleavage of the first peptide)
there should be other signal peptides or at least variants that allow import to the intermembrande space and the inner membrane (via the TIC23 complex). [I couldn't find any sources for this right now]


Sources I could find: 


TIC/TOC complex in wikipedia  
in-depth scientific review