Unity

Drop the FBX into Assets

Copy the downloaded FBX into your project's Assets folder. Unity imports it as a model with one animation clip inside. The skeleton in the file is your character's, with your bone names, so the clip plays on your character directly.

Rig tab: Humanoid or Generic

Select the FBX, open the Rig tab in the Inspector. Choose Humanoid if your character is a Humanoid in Unity - set Avatar Definition to Copy From Other Avatar and pick your character's avatar, so the clip is mapped exactly like the character - or Create From This Model, which works because the bones are named and posed the way Unity's mapper expects.

Choose Generic if you want the exact bone rotations with no Humanoid remapping in between - for stylised characters or when the character itself is Generic. Set Root node to the hips bone.

Click Apply.

Animation tab: root motion and looping

Open the Animation tab. Travelling clips move the root; to have the animation move the character, leave Root Transform Position (XZ) un-baked and enable Apply Root Motion on the Animator component. To keep the character in place and move it yourself, tick Bake Into Pose on Root Transform Position (XZ) and Root Transform Rotation.

For clips generated with the loop option, tick Loop Time (and Loop Pose); the first and last frames match, so the seam is clean. One-shot clips such as a sit-down should stay un-looped.

Click Apply, then drag the clip into your Animator Controller.

Scale

Unity works in metres, Y-up. If the character was uploaded in centimetres (typical for FBX from Maya or Unreal) and the clip imports 100 times too large or small, set Scale Factor on the Model tab to 0.01 or 100 and Apply. Convert Units usually handles this on its own.

Mixamo characters

A Mixamo-rigged character (mixamorig:Hips ...) is recognised on upload, and the animation you download uses the same names. Import it exactly as you would a Mixamo animation: Humanoid rig, avatar copied from the character.