They said to say . . . It usually means to tell others. For instance, in the frequently used Vayidabeir Hashem el Moshe leimor, G-d said to Moshe to say leimor means to say to the Children of Israel. The Kedushat Levi answers this question through answering another question. The People of Israel were happy over the salvation at the Red Sea.
Happiness is essentially an inner emotional experience, in the heart.
The answer is based on an emotional reality. Inner happiness can be fleeting and end after a short period of time. The people of Israel wanted to hold on to their joy and celebrate with G-d for an extended period of time, on and on. Speech, verbalization, and singing holds on to joy, expands it, cultivates it and enlarges it. They sang so they could stay joyous even longer and prolong their joyous love of G-d after the miracle he did for them.