The omer connects Pesach and Shavuot. Between the Exodus and Sinai we were fed with manna.
The omer reminds us of the manna. In both cases there is a set portion. With the manna, it was impossible to hog it, because no matter how much we took we had the same amount in the end. With the harvest, it is our responsibility to not hog it – we must leave over some grain in our fields for the poor.
The omer offering must be brought before we can enjoy the new harvest. It reminds us that G-d took care of us with the manna, and that G-d continues to take care of us with each harvest. The harvest is not “ours”, even though we work the land. We didn’t make it. G-d did. And we have a responsibility to regulate our portions and leave some behind for the poor.
In general, the omer, along with berachot for food, etc, reminds us that G-d sustains us.
The connection with Pesach and Shavuot is that it reminds us that G-d redeemed us from Egypt. For what purpose? To serve God, which is why we are led to Sinai to receive the Torah. The omer is the link. It represents the sustenance that brought us from Egypt to Sinai, and it reminds us that God gives us food that we should use for proper purposes, ultimately to lead us and others to Torah.