An omer is a measure of grain, in this case, barley. In the days of Temple worship, we were to bring the first sheaf of barley to the Temple, followed by other items, as a sacrifice (Leviticus 9–14).
The mechanism to connect the Exodus with the giving of the Torah is a simple one-counting the days.
Jewish communities around the world marked the "new year for the trees" last week with tree planting ceremonies and seders that celebrate Israel's seven species (wheat, barley, grapes, figs, pomegranates, olives and dates if you are keeping track!).