When loading a graphic for a FlxSprite, you can specify is as an animated graphic. Then, using animation, you can setup animations and play them.

// sprite's graphic will be loaded from 'path/to/image.png' and is set to allow animations.
sprite.loadGraphic('path/to/image/png', true);

// add an animation named 'run' to sprite, using the specified frames
sprite.animation.add('run', [0, 1, 2, 1]);

// play the 'run' animation
sprite.animation.play('run');

In the Demonstration, we are loading this image:

Animated Sprite
This image contains 7 frames of animation for a 24x24 pixel sprite.

Demonstration

package;

import flixel.FlxG;
import flixel.FlxSprite;
import flixel.FlxState;

class PlayState extends FlxState
{
	override function create()
	{
		super.create();
		
		var sprite = new FlxSprite();
		sprite.loadGraphic("assets/anim-sprite.png", true, 24, 24);
		sprite.screenCenter();
		var animationFrames = [1, 2, 3, 4, 5, 6].concat([for (i in 0...45) 0]);
		sprite.animation.add("shine", animationFrames, 15);
		sprite.animation.play("shine");
		add(sprite);
	}
}