Let's rephrase that. It means that other people can demand you prove to them how you voted, and punish you if you didn't vote the way they wanted you to.
Having a public record of the vote is only acceptable if you are permitted to forge it freely, and no record of that forgery is kept. Then you could vote one way and tell your coercer that you voted the other, and the coercer would be none the wiser. (This only works as long as the ballot is secret too, of course.)
This sort of thing is why a lot of countries make it a criminal offence to take photographs inside the voting booth.
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