The extreme case is if everyone is on solar and it's a sunny day. Everyone is trying to dump power into the grid, but there's no where for it to go. That's when you'll start causing overloads.
On those days, everyone will also be trying to run their air conditioning full blast, and although newer homes will be adding power to the grid, it probably won't balance out the extra usage from all the older, less insulated homes and businesses.
Besides, unless I'm misremembering my basic electronics, having extra power available is usually not a problem unless there is someone to consume it (*). I can hook up one side of a 110 volt outlet to a piece of aluminum foil, and until someone is stupid enough to touch it, it won't burn up. Overloads are caused by demand exceeding the available supply as it passes through some resistance (the wiring, for example). If all the houses are producing way more power than they need, that's not a problem, because the current isn't flowing anywhere. It becomes a problem when some business that normally draws power through some massive feeder lines from a cogen plant starts drawing power from all of those houses through wires that weren't designed to allow that much current draw.
Basically, the utility companies are mad because for the most part, they used to be able to ignore residential usage of electricity, because it almost never involved enough power to require precise monitoring. Now that they're suddenly able to produce power that might be consumed elsewhere, the wiring has to actually be big enough to potentially carry all the current that their rooftop systems might produce, and that requires a little bit more safety planning, and in some cases, limiting the number of solar installations and/or increasing the size of wires and transformers.
(*) There is an exception to this rule. When you have mechanical generators, having excess power is bad, because the generators have to run within a certain speed range, both to prevent damage to the generators themselves and to stay in phase. If the draw is too low (or too high) for the amount of mechanical energy going in, you could have a serious problem unless the generators have built-in governors. Of course, this problem can be solved by shutting down generators that aren't needed. More importantly, power companies have to do this anyway in response to varying load throughout the day, so the presence of solar doesn't change things very much except for possibly making the fluctuations more or less frequent and/or more or less severe.