When these underground drains become clogged, the water is no longer able to drain properly from the gutter system. This will cause water to overflow in different locations, and eventually cause damages to your gutter system. Keep in mind the drain could be partially clogged and allow some water through, or it could be completely clogged which won’t allow any water through.
Downspouts are connected to the underground drains with various sizes and shapes of adapters. Or at least they are supposed to be. All these adapters provide the same function, to make the transition seamless from one size of pipe to another. They are supposed to prevent any water from leaking out of the downspouts. (Keep in mind that if you do not have the proper downspout adapter that the water will spill out every time it rains, no mater if the drain is clogged or not.)
Does your property retain water in certain lawn spots? If so, you might have yard drainage issues. Remember that water always finds its level, taking the path of least resistance and flowing downhill. If the excess water does not have a way to flow out, once the landscape reaches its water absorption limit, then it will cause the yard to flood.