Imaam Ibn al-Qayyim رحمه الله said,
"Know that the verb 'aadha and its derivatives carry the meaning of being careful and wary, guarding and fortifying, being rescued and victorious. Its essential meaning is to flee from that which you fear will harm you to that which will safeguard you from it. This is why the one you seek refuge with is named ma'aadh and malja' (the source of refuge and recourse).
In the hadeeth there occurs, 'When the daughter of al-Jawn entered upon the Prophet صلى الله عليه وسلم [after their marriage] he moved his hand [to touch her] and she said, "I take refuge with Allaah from you." He صلى الله عليه وسلم said, "Indeed you have sought refuge with the Ma'aadh, return and rejoin your family."' 
Therefore the meaning of a'oothu is: I take refuge, guard myself and take precaution. There are two opinions concerning the basis of this verb.The first is that it is derived from the meaning of as-satar, covering or protection, and the second is that it is derived from the meaning of luzoom a-mujaawara, firmly adhering to what which adjoins it.
As for the first opinion then the Arabs used to say with regards to a house that is in the shade of a tree - 'uwwadha. Therefore when this house did 'aadha with this tree by being built under its shade, the Arabs named it 'uwwadh. The same applies to the one who takes refuge, for he seeks protection and cover from his enemy with the one he seeks refuge with.
As for the second opinion, then the Arabs used to say regarding flesh that was stuck to a bone and could not be removed, uwwadha, because of its refusal to be dislodged from it. The same applies to the one taking refuge, for he sticks firmly to the one he is seeking refuge with and refuses to be distances.
Both of these opinions are correct, for seeking refuge include both. The one taking refuge seeks protection with the one he is seeking refuge with and sticks firmly to him. His heart attaches itself to him and holds firm, just as the child sticks close its father when threatened by an enemy. The same applies to the one taking refuge, for he flees from his enemy who desires his destruction to his Lord, throwing himself between His hands, holding firmly to Him, sticking close to Him and resorting to Him.
Now, know that the reality of seeking refuge that is established in the heart of the believer surpasses and is beyond these descriptions, for these serve only as examples and representations. As for that which is established in the heart in its taking refuge, holding fast to, and its throwing itself before its Lord, its need of Him and its submission and humility before Him, then all of this is beyond description. In a similar vein, love of Him and fear of Him can only be described in a deficient way, for they cannot truly be understood except through experiencing them. This is similar to the case of one trying to describe the pleasure of sexual intercourse to one who is impotent and feels no sexual urges. No matter how much you describe it and how many examples you give, never will he truly understand it. However, if you were to describe it to one who does have these urges and has had intercourse, then he will understand your description completely.
If it is asked: When one is commanded to take refuge with Allaah سبحانه و تعالى, why does the form of the command carry a seen and taa? For example, in His saying,
Yet one says, 'I take refuge,' (a'oodhu) and, 'I took refuge,' (ta'awwadhtu) without including the seen and taa?
The reply is: the seen and taa are gramatically used to denote a person's seeking something. Therefore, when one says, 'Asta'idhu with Allaah,' he is saying, 'I seek refuge with Him.' When he says, 'Astaghfirullaah,' he is saying, 'I seek the forgiveness of Allaah.' However, when the person says, 'I take refuge (a'oothu) with Allaah,' he is actually implementing and realising what he seeks because he sought refuge and protection with Allaah. There is a clear difference between actually taking refuge and seeking refuge. Therefore, because the one who is seeking refuge is actually recoursing to Allaah سبحانه و تعالى and holding firmly to Him, he says the verb that denotes this rather than saying the verb that denotes that he only seeks this.
The opposite is true for the saying, 'Astaghfirullaah,' (I seek the forgiveness of Allaah) for in this case the person is asking Allaah سبحانه و تعالى to forgive him. Therefore when he says, 'Astaghfirullaah,' he is implementing what he desires because the meaning of this statement is, 'I ask Allaah that He forgive me.'
This, then, is the best way of seeking refuge and it was for this reason that the Prophet صلى الله عليه وسلم used to say, 'I take refuge with Allaah from the accursed Shaytaan,' and, 'I take refuge with Allaah's perfect words,' and, 'I take refuge with the Might and Power of Allaah,' saying, 'a'oothurather than 'asta'idhu. Indeed, this is what Allaah سبحانه و تعالى taught him to say, with His words,
Employing the word 'a'oothu rather than 'asta'idhu."