They are at war because Hezbollah attacked... again.
Read accounts of former UN peacekeepers who've served at UNIFIL, or actual thoughts of Lebanese themselves. Israel has been longing to acquire South Lebanon since ages, and the only thing that has prevented them from doing so were the Hezbollah. Israeli troops would block roads, fire and shoot at UNIFIL positions, as well as carry out espionage and sabotage. Every peacekeeper will attest to Israeli troops being far more problematic and dangerous than Hezbollah attacks and rockets. Lebanese themselves will echo the above opinion, and further add that Israeli looting is pretty much the same as Russian looting in Ukraine - everybody stopped buying stuff because they would know that IDF troops would seize it from them within days during the previous occupation. Shops would rather remain shut and lose business than keep themselves open under Israeli watch. Even moderate Israeli media has been extremely hostile to the very idea of UNIFIL.
The Hezbollah has always been a boogie man excuse for the Israelis to expand into Lebanon. Well, Hezbollah's gone now and we already know what's happening. Lebanon is losing close to a fifth of its land.
Israel has been bombing (and conducting raids in?) Lebanon for years. They attacked Hezbollah's ally, Iran. And Hezbollah has been attacking Israel for years. It's not true that the conflict began with Hezbollah's recent actions.
No proof of course.
Let me repeat: They are about to annex a sovereign nation while reducing the capital city to rubble. May or may not remind you of another country further north.