Many Hindu gods are associated with an animal which they ride (known as their vehicle or Vahana). There is a story and symbolism attached to each vehicle of Hindu gods and goddesses. Ganesha gained his animal vehicle, the rat, after fighting the demon Gajamukha. Rather than killing the demon, Ganesha cursed him, throwing his right tusk at Gajamukha, and transforming him into a rat. He then mounted the rat, keeping it forever under his control.
The rat represents greed and uncontrollable desire. Rat nibbles on everything destroying them. Ganesha keeps a check on this aspect.
Rat is a big problem in India for farmers. In ancient times almost all people depended on farming. For a farmer rat is there biggest enemy. By making rat his vehicle Ganesha helps in keeping the rats in control. They will no longer destroy crops and grains.
There is another popular story which states that the ashram of Sage Parashara was infested with rats. The sage prayed to Ganesha to solve this problem. Ganesha then controlled the rats and made it his vehicle. From that day onwards Ganesha is depicted as riding a rat.
Rats are known to create numerous unmanageable problems for human beings. Ganesha is the remover of all problems and it is only natural for him to be called to solve this problem too.