Consider charge. In an SN1 reaction, we have a very good leaving group that just leaves, creating a carbocation. Now, we have two charges. We have a positive charge on the substrate and a negative charge on the leaving group. If we had a strong nucleophile (generally, a strong nucleophile is negatively charged) then it will be able to react with the substrate and neutralize the positive charge, but the problem arises when considering the negative charge on the leaving group. How can it neutralize its charge?
A weak nucleophile is neutral, so when it reacts with the substrate, the positive charge doesn't get neutralized. Instead, the positive charge jumps onto the newly attached nucleophile and the leaving group is able to neutralize itself (usually by pulling off a proton). This gets rid of both the negative charge on the leaving group and the positive charge on the newly attached nuc.