"With" will never start a *clause.* That would look like: "With businesses are hiring more people now than they were at the same time last year..."
A 'clause' is a *subject-verb*.
I suspect you mean 'with' with a verb-ing modifier: "With businesses hiring more people now than they were at the same time last year, the economy continues to show signs of recovery."
'with' modifiers like this are definitely tricky, and there's not a lot of firm 'rules' I can give you. Basically it means 'this phenomenon is connected to the one in the one in the main clause.'
"The long-established TV networks have taken a beating since the advent of streaming services, with live ratings now at an all-time low."
So 'all-time low ratings' is a phenomenon 'linked to' the TV networks taking a beating.