The conflict actually started with a Soviet diplomat "misinforming" them of Israel preparing to attack them or something. This resulted in Egypt shutting off access to the Red Sea at the straights of Tiran--which they promised to keep open after another conflict--and start building their forces along the border. Israel responded by mobilizing theirs, they went to war, and Israel overwhelmed and crippled them with their Air Force.

Egypt then went to the Jordanians and Syrians telling them now was the time to attack Israel. The leader of Egypt was too embarrassed to tell them he lost though, and instead lied about crippling Israels forces. So when the other two countries came in to attack, Israel had the upper hand. Here's a map of before and after the war

Naturally, these three countries then went to the UN to bitch. The same countries that caused this situation in the first place--the ones that carved up the continent with their imperialist expansions and dumped all those Jewish refugees there due to their own anti-semitism--declared Israel's claim to the land illegal.

Why was it illegal? Because the member nations which started 2 fucking world wars decided acquisition of territory through conflict was bad... after they got done determining new borders themselves.