Similar to the answer above by R. Yaakov Kamenetsy, The Baal Hatanyah explains that the physical enslavement of the Jewish people in Egypt reflected the enslavement of their souls by the Kelipah of Egyptian impurity. Their Exodus from Egypt likewise represented a spiritual liberation from this Kelipah. Since the spiritual Exodus was an act of escape — i.e., their soul broke away and “escaped” from the impurity of Egypt, while the body and animal soul were still in exile within the Kelipah — therefore the physical Exodus likewise assumed the manner of an escape.
Tanya, middle of Chapter 31:
(source - chabad.org)
והנה בחינה זו היא בחינת יציאת מצרים, שנאמר בה: כי ברח העם
This form of divine service — in which the divine soul breaks free of its exile within the body, while the body and animal soul remain in their lowly state — is analogous to the Exodus from Egypt, of which it is written that “the people escaped.”
The Jews told Pharaoh that they would leave Egypt for only three days, but upon being released from his land they escaped.
דלכאורה הוא תמוה למה היתה כזאת, וכי אילו אמרו לפרעה לשלחם חפשי לעולם, לא היה מוכרח לשלחם
At first glance it seems strange: Why should it have been so, in a manner of flight? Had they demanded of Pharaoh that he set them free forever, would he not have been forced to do so, having been stricken by the Plagues?
The explanation, the Alter Rebbe goes on to say, lies in the spiritual aspect of the Exodus, and this was reflected in its physical counterpart just as every event in Jewish history reflects a parallel spiritual process.
The corporeal enslavement of the Jewish people in Egypt reflected the enslavement of their souls by the kelipah of Egyptian impurity. Their Exodus from Egypt likewise represented a spiritual liberation from this kelipah. Since the spiritual Exodus was an act of escape — i.e., their soul broke away and “escaped” from the impurity of Egypt, while the body and animal soul were still in exile within the kelipah — therefore the physical Exodus likewise assumed the manner of an escape.
אלא מפני שהרע שבנפשות ישראל עדיין היה בתקפו בחלל השמאלי
But escape was necessary because the evil in the [animal] souls of Israel was still strong in the left part of the heart, the seat of the animal soul,
כי לא פסקה זוהמתם עד מתן תורה
for their impurity (the impurity of kelipah) did not cease until the Giving of the Torah.
רק מגמתם וחפצם היתה לצאת נפשם האלקית מגלות הסטרא אחרא, היא טומאת מצרים, ולדבקה בו יתברך
Yet their aim and desire was that their divine soul leave the exile of the sitra achra — the impurity of Egypt, and that it cleave to G‑d.
וכדכתיב: ה׳ עוזי ומעוזי ומנוסי ביום צרה וגו׳, משגבי ומנוסי וגו׳,והוא מנוס לי וגו׳,
So it is written — that there is a divine service which consists of the divine soul’s “escape” from the impurity of the body and animal soul: “G‑d is my strength and my fortress, my refuge in the day of affliction”; “[He is] my high tower and my refuge”; and9 “He is my escape...”
And the Exodus from Egypt exemplified this idea of “escape”.
ולכן לעתיד, כשיעביר ה׳ רוח הטומאה מן הארץ, כתיב: ובמנוסה לא תלכון כי הולך לפניכם ה׳ וגו׳
Hence it is written of the Redemption which will take place in the time to come, when G‑d will remove the spirit of impurity from the earth and there will therefore be no evil necessitating spiritual escape: “[You will not go out in haste,] nor go in flight, for G‑d will go before you.”
The Exodus from Egypt, however, took place in a manner of flight, for the evil was still strong in the people’s animal soul. Similarly, whenever one disregards the lowliness of his body and animal soul and engages in the Torah and the mitzvot in order to free the divine soul from its corporeal exile, he effects the spiritual equivalent of the Exodus from Egypt.