To stay in touch is to maintain communications with someone. Other idioms having the same meaning are: keep in touch and remain in touch.
- Although my neighbour moved to a new city, we stayed in touch.
- You should write oftener than you do. We have got to stay in touch.
- I will give you my phone number so that you can stay in touch with me.
- Will you keep / stay in touch with me after moving to France?
- I don’t know his phone number or address, so I can’t stay in touch with him.