For those of you that don't know already, the CC when loaded with the proper troops is the absolute strongest defensive building in the game. For the sake of comparison, Inferno Towers are without doubt the strongest defensive tower; at Level 3, the Inferno tower in Multi-Target mode is 42 dps X 5 Rays = 210 Damage/Sec. A Level 5 CC loaded with Level 6 Archers with 22 dps x 30 = 660 Damage/sec. If you swap that with 7x Level 5 Wizards + 2 Lvl 6 Archers it's a whopping 7 x 255 dps + 2 x 42 dps = 1614 Damage/sec! That's 8 time's stronger than the Inferno Tower, and lets no forget their Range and Splash damage too! This is why Wizards and Archers are so popular as the de-facto standard of donations, they are great all-round ranged troop that can shoot from a distance, and attack both air and ground units.
Getting to the point, so a loaded CC packs a huge punch on your attackers. So what? I just keep it loaded...right? Not exactly. For farming purposes, this tactic may be enough to keep most of the regular attackers at bay. However, with high level trophy pushing that's typically above Crystal and beyond, a more elaborate strategy is required. Over at the Supercell forums, it's been well document as a trophy pushing guide to always lure out CC troops before actually dropping you're entire army. Basically to take this X-factor right out of the attack equation to guarantee a victory in battle. When attacking a base, it is taught as one of the first things you need to do is to draw away all Heroes and CC troops by getting their (!) attention by placing sacrificial troop(s) within their range. When that is done, the attacker finds an open building somewhere in your base, preferably in a open corner where then they send a range of troops to destroy your CC troops and Heroes.