Why is this sentence correct?
She suggested that he go to the cinema.
I would definitely use goes instead of go.
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For those who are a little confused by Barrie England's answer...
She suggested that he go to the cinema. and She suggested that he goes to the cinema. are both correct, but they have different meanings.
Here's how she might suggest that he goes to the cinema:
ALICE: Where do you think he goes every Thursday evening?
JANE: Hmm ... well ... cinema tickets are cheap on Thursdays, and he loves movies. Maybe he goes to the cinema?
This is similar to She said that he goes to the cinema., but with less clarity or certainty.
Here's how she might suggest that he go to the cinema:
BOB: I want to go out and have some fun this evening. Do you have any suggestions?
JANE: Why don't you go to the cinema?
This is similar to She told him to go to the cinema. but with less force.
So why do we use go rather than goes. This is an example of the use of the subjunctive mood. The subjunctive is much less important than it used to be in English grammar — many observers of the English language think it is dying -- but it is still used in phrases like prefer (that), suggest (that), vote (that), wish (that) and so on.
The subjunctive is used more in American than in British English. To me, as a native speaker of British English, She suggested that he go to the cinema. does not seem wrong (perhaps because I've seen and heard so much American English), but I might prefer to say She suggested that he should go to the cinema., which has the same meaning, but does not use the subjunctive.
Getting back to the point in a way that might help the questioner, I say that the construction in question is used for suggestions and orders. It is rather like saying should go/ ought to go/ might go/ may go / could go, but without mentioning any one of those words. It is commonly used in the law:
The court orders that: 1. the Respondent refrain from harassing the Applicant; 2. the Respondent do not approach within one hundred yards of the Applicant's home.