How do I throw a boomerang or spear in Breath of the Wild?

You can throw any weapon by holding down and releasing the upper right shoulder button (R). A thrown weapon will deal double it's base damage and stagger most enemies.

Source on double damage:

Critical Hits – Weapons that you have no use for are perfect for throwing at monsters. If one hits, it will deal double damage.

Keep in mind that throwing a weapon will take away a lot of it's durability, sometimes destroying it instantly.

Boomerangs fly back to you when thrown but you still have to catch them manually by pressing A in the correct moment.

Throwing a normal weapon is usually a good idea if there's only one hit left.

While your weapon is drawn, hold down the R button, the release it when aimed where you want to hit. This can be done with any weapon, not just spears and boomerangs.

When the boomerangs returns to you, you need to press A to actually grab it, otherwise it will simply go past you.