Ты нужен Тому здесь.

Shouldn’t this be, “You need Tom here”?

This sentence means “Tom needs you here”. Because нужен isn’t a verb but an adjective.

Literally it means: For Tom, you are needed here.

“For Tom” uses the Dative case (therefore the y at the end of Тому), and “you” is accusative.

You have a few other similar “adjective-verbs” in Russian (can, cannot, need, must, and so on).