This is typical. The alarm system is normally powered by your home AC power. It may be hardwired, though typically for small alarms use a plug in adapter with low voltage wires routed through the wall to the main alarm unit. There is also a battery backup, possibly rechargeable, but even rechargeable batteries can wear out.
When you turned off the power, the battery discharged enough to indicate the need for replacement. When you turned power back on, that allowed the alarm to work again but did not remove the low/replacement battery condition.
Battery replacement is usually pretty easy - a few screws and then two wires. The batteries can vary from a traditional 6V "lantern battery" to a modern rechargeable battery pack. While the alarm company should be able to tell you exactly what type of battery you have/need, often the easiest way to figure it out is to simply find the battery, remove it and take it to a battery store, or order a replacement online based on the battery specifications.
Many alarm companies will notify you automatically in case of a low battery situation. This is important because if the low battery happens while you are away you may not realize that there is a problem until the battery goes dead and the alarm no longer works. They may also have tried to notify you on an old number - e.g., many people no longer answer their landline except for friends/family - so periodic updating of the alarm company notification information is worthwhile too.