Rav Menachem Mendel of Rimanov zy”a explained the meaning of “Yachatz” with the following moshol:
There once were two good friends who were extremely close to each other. When a time came that they had to depart from each other, they wondered how they could keep their bond of love and remember their closeness. They concluded that the best way to do this was for them to each take a picture of their friend, which they could keep with them at all times.
What did they do? Each one had a picture of himself. They cut both pictures in half, and each took one half of each picture. Thus, each one now had half a picture of himself and half a picture of his friend.
This is what we do by Yachatz. We divide the matzoh to remind us that we will remain connected to Hashem and He will remain connected to us.
