When learning everyday expressions, my advice is that you learn them as they are, without questioning the grammar too much. If you are a beginner, it's better that you just learn the equivalences, since the explanations for some uses of language may be complicated, and will remain unclear until you reach the actual level where these structures are learnt.
That said, since, as a student, I always hated teachers refusing to explain complicated things at lower levels of learning, I'll try a simplified answer:
"Hay" is the impersonal form of the verb "haber", and it it means "there is/are" in English, and it's used to describe the things that exist in a place or context.
- En la mesa hay un libro.
There is a book on the table.
- En mi cartera no hay dinero.
In my wallet there is no money.
- No hay problema.
No problem.
- En la vida hay muchas dificultades.
In life there are many difficulties.
- En el nido hay tres pájaros.
There are three birds in the nest.
As for the sentence "¿Qué hay de ti?", it literally means "what is there about you?"
The question is asking for new information from you, but not in a weird way, only wanting to know how have you been lately or asking for the equivalent of the information recently provided by your conversation partner.
-¡Hola! Hace mucho tiempo que no te veo. ¿Qué hay de ti?
Hello! It's been a long time I haven't seen you. How have you been?/What's new?
-Trabajo en una zapatería. ¿Y qué hay de ti? ¿A qué te dedicas?
I work in a shoeshop. And what about you? What do you do for a living?
-Estoy aquí porque vine a recoger un paquete. ¿Qué hay de ti? ¿Tú a qué viniste?
I'm here because I came to pick up a package. How about you? Why did you come here?
-Me gusta jugar futbol. ¿Qué hay de ti? ¿Practicas algún deporte?
I like playing football. How about you? Do you practice any sport?