How can I repair anvils?

You cannot repair an Anvil. The Anvil will break if it's done. You can just build another one and place it on the old place of the Anvil.

You just can repair Items at the Anvil. This is the thing you see in the release logs called "Anvil repairing has been rebalanced" as it was way too expensive in older versions.

You cannot repair an Anvil, even using another one.

The main change of version 1.8 is about the prior work penalty.

You can repair an anvil, just have the almost broken one fall in lazy chunks, then it will end up as a repaired anvil. This is a glitch though.

One way to do this is to set up a redstone line that is longer that your render distance, such that the end of the redstone line should be barely visible (if at all) if you stand at one end, and thus will be loaded as a redstone-only (lazy) chunk. Remember to extend the redstone signal with repeaters where necessary, and make sure the end of the redstone line is not in spawn chunks.

Power the redstone line from one end, then place a piston facing sideways on the other. Place the broken anvil on the extended piston arm, then travel to the control end of the redstone line and depower the redstone line. The piston should retract in lazy chunks and thus the anvil should fall in lazy chunks and be repaired.