Sometimes rain gutters are run to underground dry wells. Basically a hole lined with a tub filled with gravel that allows the water to drain down into the soil. The ones I've seen have a popup overflow.
I've never seen one that has holes like yours. My guess would be either someone didn't want to run the underground pipes and just put a drain cap. Or maybe that area of your lawn is lower than other areas and pools with water when it rains so they put a drywell with a drain there to get the water down into the soil so it doesn't get muddy. Or I could be wrong and it's a drywell connected to the downspouts with holes instead of a popup.
It's most likely some sort of drywell but whether it's for your gutters or for drainage due to grading problems it's hard to tell without seeing photos that show it in context.