In plain words
Vināyaka is the giver of learning. He is the leader. In the world there are many who stand as an ideal, but there is none to equal Vināyaka. Another name of His is Vighneśvara. He is the master of obstacles. While birth and death come about by the grace of Yama, obstacles fall away by the grace of Gaṇeśa. Gaṇapati means the master of the gaṇas. He is the master of sounds. He is the embodiment of the Praṇava. He is the embodiment of Oṃkāra.
Gaṇapati has an elephant's head. What is the inner meaning of this? The elephant's head is the sign of surpassing intelligence. He is of noble Buddhi, and so He grants Buddhi. That is why He has another name, Buddhi Vināyaka. Because He grants the wishes we ask of Him, He has another name, Siddhi Vināyaka.
Once upon a time Vyāsa resolved to write the Mahābhārata. But it was a very hard task. For it held a hundred crore ślokas. Vyāsa prayed to Bṛhaspati, the preceptor of the gods. When Bṛhaspati told him, “Except Vināyaka, there is no one else who can help you,” Vedavyāsa at once prayed to Vināyaka. That very moment Vināyaka appeared before him.
Vyāsa laid down a condition: there must be meaning in every word that is written; it must be written down only after it has been understood; and there must be harmony between one word and the next. Gaṇapati too laid down a condition: that he must dictate without a pause, without stopping for breath, and that His stylus (the pen of those days) must not stop. Because Vināyaka is the embodiment of Jñāna, He understood the Mahābhārata completely and wrote it down. That is why Vyāsa was able to write the Mahābhārata with His help.