What's this?
...reduce third party registration...”
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It is not talking about you registering to be a member of a third party.
It is talking about how third party groups such as ACORN and ACLU etc. organize and register voters.
This is to put out of business the fraudulent voter registration groups like ACORN, SEIU, etc., etc., and to limit the dead people, incarcerated people, and illegals that magically show up as registered Democrats, and magically get their Democrat votes counted.
People used to have to expend some effort and brainpower to register to vote, and then to vote.