Manh Nguyen's link correctly explains that this is a use of the present subjunctive. That article also mentions that the subjunctive is dying out in English. If you want to be extra-correct, such as when doing a test, the subjunctive form is the one to select. Teachers love to jump on students who don't!
However, in everyday usage it's better to use the indicative form. So it's quite ok to say, 'He prefers that she speaks with him personally.'
Beware the danger of overly formal language, which is just as incorrect as language that is sloppy. Language always occurs in a context, and it is one of the great skills of native speakers (of whatever language) that they can adapt what they say to the situation.
In my experience, most learners of English err on the side of being too formal. That's certainly applies to the subjunctive form if it is used in an inappropriate context.