It shows you are not an illegal immigrant, therefore employer is ensuring he is not breaking the law on employing you. It means you are registered on Government records as a potentially employable person. If employer needs to contact tax office about you, say to check code number, it means he can get information quickly. Tax office, however, will not give him any personal information about you or your tax/ NIC, so he gains no information about your personal circumstances. Tax office uses your NI number as a reference number to access your records. Employer is expected to quote your number on his annual employer's return, or in any communication he has about you with tax office.

It establishes you are bona fide, entitled to seek employment , not an illegal unregistered worker , or immigrant

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They want to be sure you are eligible to work here. They cannot collect any info from it at all.

So they can verify you are legally-allowed to work in the UK. It makes it easier for them than doing the same with just your name, address & date of birth.

(If you are somehow under the impression that another person armed with your name, address and date of birth CANNOT find out just about anything they want to about you & your history, you are VERY naive!)

You and your employer are legally bound to pay NI, this is also used to get your tax records, this isn't just some employers it's all of them, if you haven't paid enough NI for some reason they could be liable. this is also a 'Unique Key' identifying just one person, this stops you from being confused with another person with the same name.

Your potential employer also gets what rights you have to work in the UK, something else they are legally obliged to check prior to employment, the fines they could face if they don't check this 'right to work' status are large.