In plain words
Vyāsa telling it, Vināyaka writing it. In that way the Mahābhārata was written. In it is the essence of all the Vedas. In olden days the great sages used to do tapas. Some would give it up halfway. Some were able to complete it. When tamas is removed, Mokṣa is attained. Gaṇapati, the master of the gaṇas, wrote the Mahābhārata because He is Vācaspati.
The elephant is the symbol of intelligence. The mouse is Gaṇapati's vehicle. It is very quick. That is, the meaning is that we must do all our work in good order and with alertness. Though all the beasts roam the forest, the footprints of all the animals are lost in the elephant's footprint. If an elephant once walks ahead through a fearsome forest, a path is made. So, if we go with Gaṇapati in front of us, our walk through the forest of life goes on in comfort.
On the day of Vināyaka Caturthī, students place their books before the temple and pray to God, asking Him to grant them Buddhi. That is why He has the name Vidyādāyaka. We pray to Vināyaka so that no obstacle comes to us in any of our work. Seeing His elephant's head and His big belly, you must not think Him misshapen. His Bhakti is very great. Vināyaka is the one who showed that the whole world is within Him.
He is Śuklāmbaradhara. That is, the one who wears white garments. He is beautiful with vibhūti. He has four arms. He always gives darśan with a serene face. His wearing white garments shows that His heart is steady and spotless. If we too become broad of heart, we can win His grace. The vibhūti tells us that everything in the world is impermanent. It shows that the principle of the Ātma alone is permanent. Where man has two arms, Vināyaka has four arms. Two of His hands are always there to help mankind; that is why they are not seen.
He always has an even mind. Sorrows come and go. Joys come and go. But we must have an even mind, as Vināyaka has.