The C rating of the battery is related to how much current it can deliver. For 1000mAh 25C you approximately get 25A of burst current
If your copter drains more current than battery is able to deliver, then you put stress on your batteries.
In such condition battery will puff, because of oxidation process that also increases battery internal resistance further decreasing it's C rating and making situation even worst.
Draining too much current is like shorting the battery. It won't last long. Puffed/degraded batteries are dangerous to use, since they over-heat due to high internal resistance and building up pressure make them likely to explode and start a fire. So it's not only less power and slower charging but also risky and short living battery. Of course higher C batteries are more expensive.
When you buy battery usually C ratings are provided for charging and discharging. For discharging two values can be provided lower continuous and higher burst. If one discharge C value is provided its usually burst. Usually reported rating on the battery is higher than what battery can deliver in reality, even in high quality LiPos.