shutter count is a counter that counts how many times the shutter has been pressed since it left the factory, Depends on the model, the expected life of the shutter mechanism is around 100,000 to 150,000. But that does not means after 100,000 actuation, the shutter will fail. It is just an estimate similar to the hard disk MTBF (mean time between failure). When the shutter mechanism fails, you will need to get it replace from the service centre.
The A300 is an older model, but still a very good camera. If you can stretch your busget a little bit more, a second-hand A500 or even A550 would be better.
The shutter count is, essentially, how many pictures the camera has already taken. You should just keep on shooting, don't worry. The only time you need to bring it to the service centre is when it stops working.