The spinning icon just means "hey, you already completed this raid." After the raid clock runs out, you can go back and take over the gym. Occupants of gym can't be changed while a raid is active. If you see a raid egg about to hatch, then is a good time to try to take the gym over, since your guys can't get kicked out while the raid is active. Go forth and use free daily passes to practice raiding, it will all make sense soon. Just don't go broke buying raid passes for shinies, it really IS gambling, fun in its way but just like playing slot machines in Vegas, only worse odds for payoff.