As a complement to other answers, and especially to nbubis' answer.
Based to a large extent on the work of the Israeli archeologist Israel Finkelstein (a youtube video with him talking on this topic HERE; other similar videos are easy to find) the general problem of advancing from "biblical archeology" (archeology that aims at proving that biblical stories are describing historical facts) to archeology as such
is rather well presented in the book Oltre la Bibbia. Storia antica di Israele (2003) by Mario Liverani (Israel's History and the History of Israel), which I have read in a French translation, La Bible et l'invention de l'histoire, Bayard 2008, Gallimard. More works by Liverani: here.
Here are a few pages (Part II, chapter 14, section 4, from page 278) that would illustrate the main argument that could explain how the story may have been elaborated, what the history of the text may be.
As for the probability of it being a true fact, the other answer has mentioned decisive proof against it. I would only repeat that Canaan was occupied by Egyptian forces before the possible date of the Exodus until centuries after that. The Exodus would have taken place inside the Egyptian empire, Canaan wouldn't have been a safe place, it wouldn't have been "out of Egypt" anyway, Egyptian garrisons would have met the refugees upon their arrival in Canaan etc. Egyptian ideological and cultural influence, including Egyptian origin of some names, appears normal in this context. The map of the area is that of the Egyptian empire.
I have highlighted some important terms.
(I had initially posted my own translation from French. Thank you @LаngLаngС for the official English version!)
In these eighth-century formulations, the motif of arrival from Egypt was therefore quite well known, but especially as a metaphor of liberation from a foreign power. The basic idea was that Yahweh had delivered Israel from Egyptian power and had given them control – with full autonomy – of the land where they already lived. There was an agreed ‘memory’ of the major political phenomenon that had marked the transition from submission to Egypt in the Late Bronze Age to autonomy in Iron Age I.
We should bear in mind that the terminology of ‘bringing out’ and ‘bringing back’, ‘sending out’ and ‘sending in’, the so-called ‘code of movement’, so evident in Hosea, had already been applied in the Late Bronze Age texts to indicate a shifting of sovereignty, without implying any physical displacement of the people concerned, but only a shift of the political border. Thus, to take one example, the Hittite king Shuppiluliuma describes his conquest of central Syria in the following way:
I also brought the city of Qatna, together with its belongings and possessions, to Hatti… I plundered all of these lands in one year and brought them [literally: ‘I made them enter’] to Hatti (HDT 39-40; cf. ANET, 318).
And here is another example, from an Amarna letter:
All the (rebellious) towns that I have mentioned to my Lord, my Lord knows if they went back! From the day of the departure of the troops of the king my Lord, they have all become hostile (EA 169, from Byblos).
Egyptian texts also describe territorial conquest in terms of the capture of its population, even if in fact the submitted people remain in their place. This is an idiomatic use of the code of movement (go in/go out) to describe a change in political dependence.
But when, towards the end of the eighth century, the Assyrian policy of deportation began (with the physical, migratory displacement of subdued peoples), then the (metaphorical) exodus from Egypt was read in parallel with the (real) movement from Israel of groups of refuges from the north to the kingdom of Judah (Hos. 11.11). The inevitable ambiguity of the metaphor of movement gave way to a ‘going out’ which was unambiguously migratory, though it maintained its moral-political sense of ‘liberation from oppression’. The first appearance of this motif occurs, significantly, in the Northern kingdom under Assyrian domination.
Thus in the seventh century the so-called exodus motif took shape in proto-Deuteronomistic historiography. The expression ‘I (= Yahweh) brought you out from Egypt to let you dwell in this land that I gave to you’ (and similar expressions) became frequent, as if alluding to a well-known concept. Evidently this motif, influenced by the new climate of Assyrian cross-deportations, and the sight of whole populations moving from one territory to another, was now connected to the patriarchal stories of pastoral transhumance between Sinai and the Nile Delta, to stories of forced labour of groups of habiru (‘pr.w) in the building activities of the Ramessides, and to the more recent movements of refugees between Judah and Egypt: such movement was therefore no longer understood as a metaphor, but as an allusion to an actual ‘founding’ event: a real ‘exodus’, literally from Egypt.
Just as in Hosea the Exodus motif already provided a metaphor for the Assyrian threat, so in prophetic texts of the exilic age the exodus became (more consistently) a prefiguration of the return from the Diaspora – at first, fleetingly, from the Assyrian, to a (still independent) Jerusalem; then firmly, from the Babylonian diaspora:
Therefore, the days are surely coming, says Yahweh, when it shall no longer be said, ‘As Yahweh lives who brought the people of Israel up out of the land of Egypt,’ but ‘As Yahweh lives who brought out and led the offspring of the house of Israel out of the land of the north and out of all the lands where he had driven them.’ Then they shall live in their own land’ (Jer. 23.7- 8; 16.14-15).
At the conclusion of the whole process, in the sixth–fifth centuries the entire story of exodus and conquest of Canaan had been re-elaborated in the light of the real events of Babylonian deportation and return of exiles, thus in effect a ‘new exodus’, prefigured by the mythical one.
What I also found very interesting in this book was the argument against the actual historicity of the massacres against Canaanite peoples that the Bible boasts about (herem). There is no place here anymore for more details about that, but to put it shortly: the real peoples of Canaan that existed and continued to exist there were conveniently excepted from that harsh treatment; the ones that were put to sword are imaginary or lived hundreds of years before. (The "motif" is maybe inspired more by Mesopotamian/Assyrian ideology — than by the Egyptian.)
Among the presentations linked in the other answer there is one by I. Finkelstein, but the one by Robert Mullins seems to me even closer to Liverani's book. The map above is from that presentation (6:37).