"For" is referring to "how long." It could be reworded like this: "For how long do we look at our mobile phones in a day?" If you wanted to get rid of "for," you could say "How much time do we spend looking at our mobile phones in a day.

For is used as either a conjunction or a preposition. It represents a formal part of everyday sentence structure when .. for [fôr, fər] PREPOSITION in support of or in favor of (a person or policy): "they voted for independence in a referendum" affecting, with regard to, or in respect of (someone or something): "she is responsible for the efficient running of their department" · [more] on behalf of or to the benefit of (someone or something): "these parents aren't speaking for everyone" having (the thing mentioned) as a purpose or function: "she is searching for enlightenment" · [more] CONJUNCTION because; since: "he felt guilty, for he knew that he bore a share of responsibility for Fanny's death"

Basically, this is an example of native English speakers not using particularly good grammar. Really it should say "For how long do we look at our mobile phones in a day?" Especially if you are writing for a news outlet like BBC. Also, answering your question, 'for' isn't necessarily needed in this sentence, but if I were talking, I would definitely use it. I hope that helps!