You say that someone "has had to" do something when you're talking about a situation that started in the past and is still continuing now. For example:
I've had to work late every night this week.
Someone might say this on Thursday or Friday because the week hasn't ended yet. But you wouldn't say this if it's Saturday and you have the day off. In that situation, you'd say:
I had to work late every night this week.
In the example at top, the speaker says "we've had to turn away business" because the company is still very busy and might have to turn away business again.