What's the point of fighting monsters in Zelda BotW?
It's certainly possible to play through the game ignoring most of the monsters you encounter. However, there are certainly good rewards that can come from clearing them out.
Many monster camps are guarding a monster chest that contains a weapon or other rare item. The chest only unlocks once all monsters in the camp are defeated.
Even if the camp isn't guarding a chest, they usually have other resources nearby that are difficult to gather while monsters chase you. These resources include things like fish in nearby streams, ore deposits, or a set of weapons.
Sometimes the monsters are just annoying. I can't tell you how many times I stood still to enjoy the view or to get my bearings when suddenly I get hit in the face with a random Octorok spitball. Or a Lizalfos decided to chase me while I was chasing a Korok.
Sometimes killing monsters will save a nearby traveler. They'll thank you and give you a reward - usually some mid-to-high tier food. (Occasionally it will actually be a trap, and you'll be attacked by the Yiga clan instead... stay on your toes!)
Honestly, it just feels really good to beat the snot out of Ganon's minions. With the right combination of weapons, wiping out a whole camp can be super satisfying.
In addition to all the reasons other people have already mentioned, IMO there is an even more important reason:
The entire difficulty scaling system in Breath of the Wild is based on enemy kills.
Difficulty scaling?
Difficulty scaling is the game mechanic that results in enemies and weapons being progressively replaced by more powerful variants.
It's why some Red enemies become Blue, then Black, then Silver over the course of a playthrough.
It is also why most weapons you can find in treasure chests and in Hyrule receive upgrades and modifiers (such as Durability Up).
The scaling system is based on a point system. Killing enemies is the only way to receive points.
If you don't kill enemies, you'll just keep finding low level enemies and weapons that will never upgrade.
For more details, see: https://zeldamods.org/wiki/Difficulty_scaling
You need their drops for armour upgrades. While potion ingredients are a secondary, rather minor boost, you need to kill a lot of monsters in order to get the best armour.
- Bokoblin (78) - Hylian, Radiant, Soldier,
- Moblin (27) - Flamebreaker, Radiant, Soldier
- Lizalfos (174) - Desert Voe, Rubber, Snowquill, Soldier, Zora
- Keese (116) - Climbing, Desert Voe, Snowquill, Soldier
- Chuchu (87) - Desert Voe, Rubber, Snowquill, Soldier
- Octorok (5) - Snow Boots
- Guardian (15) - Ancient
- Hinox (34) - Fierce Deity, Soldier
- Moldulga (17) - Radiant, Sand Boots
- Lynel (33) - Barbarian, Fierce Deity, Radiant, Soldier
The number is based on rarest drop required. You may need more kills if you don't get enough of the other drops (esp. guardians). Note that the base tier of most monsters do not drop it so you'll probably still need to kill more than listed.
Silver/white tier also drop gems, which again help for upgrades and also selling for cash. You need to kill enough lower tiers for them to start spawning.
Once you've got all that then there's no need to keep engaging. Indeed, items become available (purchased indirectly with more monster parts) that make the main monsters (bokoblins, moblins, lizalfos and lynels) ignore you so you can more easily skip them.