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The answer is a combination of military and political reasons.
Israel had survived several wars up to this point, from enemies who outnumbered and wanted to completely destroy them. Their success had created a strong belief in Israeli might, both internally and externally. This war changed their outlook.
Israel had caused severe damage to its enemies and had occupied the Sinai Peninsula. A continuation of the war would have cost Egypt dearly to recapture it. However the war was not totally successful for Israel. They had an army in Egypt that was surrounded and would have taken heavy casualties to be extracted (Politically catastrophic as Israel has always been sensitive to casualties due to their low population). The Israeli cost of the war was high due to being caught unaware. Therefore, both parties wanted a way out of the war with the least amount of damage.
From a real politik perspective, if Egypt has failed to destroy Israel and conceivably would continue to fail, it would be more beneficial to acknowledge the current state of affairs and act accordingly rather than beat its head against a wall over and over. Israel would have been more eager to seek a peace, as the war had shaken their psyche, and having lost their "invincible" status they anticipated further attacks from previously beaten enemies.
In exchange for returning Sinai, Egypt acknowledged Israel's existence. I believe there was a political reason, in terms of Egypt was wary of another Arab power (probably Saudi Arabia).
Economically, Egypt made the transition from the Soviet bloc to the Anglo bloc. This realignment with the Americans brought large amounts of Economic and Military aid, as well as diplomatic and trade concessions with the members of the American Sphere. The thawing of relations was very beneficial to both parties. Economically for Egypt, and militarily for Israel.
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Question: Why did Anwar Sadat establish diplomatic relationship with Israel and did that decision pay off?
Timeline
Full Normalization of Relations, including recognition and exchange of ambassadors, were part of the terms of the 1979 Egyptian Israeli Peace Agreement. Why did Sadat agree to the peace plan. The condensed answer is it was in Egypts best interest to do so and Israel gave up significant sacrifices to secure the recognition and promise of peace. Israel gave up the Sinai Peninsula and all their territorial gains from the last two wars with Egypt. Egypt broke with the Arab league in recognizing Israel and that and long term peace were what both sixes hoped to achieve. Was it in Egypt's and Israel's best interest, this could be interpreted as an opinion question. My opinion is it was worth it.
On the Negative Side.... From Israel's perspective they never got warm relations with Egypt, never got the economic relationship Israel desired. From Egypt's perspective Sadat was assassinated, that was a loss. Egypt was also ostracized from the Arab world for a time.
On the Positive Side... Israel lost her largest and most existential threat on her boarders, Israel gained use of the Suez Canal, unrestricted maritime access to their Red Sea Port, and recognition from an important country in the Arab World.
Egypt as I said got all her territory back she lost in the 67 and 73 wars which was an Egyptian demand which Israel struggled with but ultimately agreed too. Keeping a military buffer against future Egyptian aggressions was a vital Israeli concern.
Finally and most importantly there has been no war between Israel and Egypt since the peace agreement, and that's significant because in the three decades leading up to the 79 Peace Treaty they fought 4 major Hot wars and series of multi year warm conflicts. Ultimately I think that makes it worth it for both sides.