Two ways assuming the work is owned by someone else:
1) Remove the item that is causing the copyright issue.
2) Get written agreement from the owner of the copyrighted material that allows you to use that material. Sometimes that means payment (royalties), sometimes not.
One way – assuming the copyright claim is not valid:
Another way has you defending your use of the copyright claim not being accurate. For example, I had an invalid copyright claim on music – I knew it was invalid because I wrote the music. I pointed out in my appeal email to YouTube that this was not the copyrighted material that was claimed, they removed the copyright claim.
There are instructions about dealing with copyrighted material at the YouTube support area.
If you are sure the claim is wrong and contents are yours or you got permission to use it then you could do as follows:
Contact the claimant provide details why they should retract the claim.
Counter file claim with YouTube and provide details which should proof contents are yours or you got permission to use them. In both cases be polite. It may take time to get reply.
Note do not file counter claim unless you are sure contents are yours or you got permission to use.
Negotiate a licensing deal with people who own the copyrights to the work you are using illegally.
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remove the copyrighted material from the video and reupload it.