I was wondering how I could create this effect in blender for an animation:
How can I make this bloom effect?
When dealing with these effects, generally you have 2 ways - create some geometry with high amount of lighting and use volumetrics for them or fake these effects with compositor.
Using volumetric materials
You should have some geometry providing actual rays of light; this may be some basic mesh which has forms of rays, or particles emitted from somewhere, forming cloud of light. Then with using volume materials you'll get desired effect.
To begin, either model basic form of rays as mesh as @PGmath suggested or create particles system on the object, which is supposed to emit light, and render those particles as volumetric halos with using Point Density texture which is what I did. See What is Point Density for more info about this feature. See Halo particles don't render in Cycles for details about texture setup (Note: accessible starting from Blender 2.76).
Create several cubes which will contain particles inside. They will serve as domains for volumetric materials. I added particles coming from the floor too, for adding a bit of volumetric lighting across the scene.
Also you may want to position illuminating objects on different layers; this could improve scene allowing to:
In material setup for the domains add Texture > Point Density and Shader > Emission shader which will be connected to Volume output socket of Material Output node. Set Point Density to Particle System mode and choose adjacent one. You may want to add Math node in order to control amount of final effect.
Play animation to let particles emit. After rendering and a bit of post-processing you'll get something like glow where particles are located.
Faking effect in compositor
While you will get nice result with volumetric material it's very heavy for rendering. And since you asked about animation the rendering times may be even more important. Hence it may be worth trying to fake it with post-processing (at least, part of effect). From this point of view, I think How to render lightning in Cycles is related with only exception for actual object.
In case of this scene, I used following steps:
In Composite nodes with keying get part of image which is going to be heavily processed. Create a mask which will control where post-processing will occur. Overlay them together.
The Post-processing group is the one used in the first way and it was used for refining the result (adding Glare, Sunbeams, vignette, desaturation of some parts of image and Lens Distortion).
The rendering is ~ 15 times less.
There's a lot of room for improving:
It may depend on the actual scene requirements / limitations.
I used this setup with a bunch of cones and a Volume Emission material You can parent them to an empty and scale it with noise, to create flickering effect. Or even better parrent to few Empties and create separate flickering effect