Israel launched what it said was a preemptive strike [0] against Egypt, destroying virtually the entire Egyptian air force on the ground. Lacking air cover, Egyptian ground forces in the Sinai peninsula retreated, but were destroyed by the Israeli air force on the roads.
Not having a clear picture of what was going on [1], the Jordanians and Syrians entered the war on Egypt's side, only to suffer a similar fate as Egypt.
Israel ended up conquering the Sinai peninsula, East Jerusalem and the West Bank, and the Golan Heights.
It's hotly debated whether the war really was a preemptive strike to head off an impending Arab invasion, or whether it was a war of aggression by Israel.
In any case, as a consequence of the war, millions of Palestinians came under Israeli rule. The Israelis were faced with a conundrum: they wanted to keep the land (especially the West Bank and East Jerusalem), but they didn't want millions more Arab citizens, who would shift the demographic balance, ending Israel as a Jewish state. That's why Israel has kept the Palestinians in limbo for more than half a century. They live under Israeli control, but they aren't Israeli citizens.
0. Israel initially claimed that it had been attacked first, but it soon admitted that it had attacked first.
1. Egyptian propaganda in the first hours claimed that Egypt had scored major victories.