Infinitive without to

• *You need stay at home is an ungrammatical sentence.

Need, like dare, is a semi-modal verb. That means it can act like a Modal Auxiliary verb in certain situations, of which this is not one. When it does act like a modal auxiliary verb, it can take an infinitive without to; that's what modal auxiliary verbs like must, can, and would do:

• You must stay at home; she can stay at home; I would stay at home.

So, when do semi-modal verbs get to act like modals?
In negative contexts. Semi-modality is a Negative Polarity Item, like ever or budge

• He sat down and didn't budge for an hour, but not *He budged for a while
• He doesn't ever go down there, but not *He ever goes down there
• You need not stay at home, but not *You need stay at home
• He dare not tell the story, but not *He dare tell the story

When semi-modals don't occur in a negative environment, they just take infinitives with to, like most verbs; it's only with negation that they do tricks.