If your opponent has no monsters, and you have one powerful monster, you can activate "Shien's Spy", and then Summon this card, coping the stats of your own card. For better results, activate "Owner's Seal" to bring back your monster before the End Phase, and have two identical monsters.
If your opponent has a powerful monster and you need to get rid of it, use "Kishido Spirit" and then Summon this card, copy the ATK of that monster and then attack it. This way, you get rid of your opponent's monster and get a powerful monster on your side.
This card works well with "Junk Warrior". By copying a high ATK monster on your opponent's field, "Junk Warrior" gains ATK equal to the ATK copied by Copycat.
This card can attack directly if you used with "Baby Tiragon" and can do a lot of damage if your opponent has a strong monster. As a bonus, this card is Level 1, and can be used to Xyz Summon Baby Tiragon (or another Rank 1 monster) if used in the same deck.
This Card can be special summoned by the effect of "Magical Exemplar" (by removing just 1 spell counter; each time a spell card is activated put two spell counters on Magical Exemplar)
This could be used in a Piper Chaos deck, by Summoning this with "Kinka-byo", you can potentially have at any moment in the duel, something to destroy your opponent's threatening monsters with. By attacking a monster with equal attack, this will go back to the grave for re-use instead of being banished.
If your opponent has a monster with 2000 or more ATK out on the field, and you already have one with 2000 ATK or more, this card can be useful for Special Summoning "Galaxy-Eyes Photon Dragon".
If your opponent has only face-down monsters on the field and you have at least one monster with decent attack, summon this card. Since there are no face-up monsters, its attack and defense will remain zero. Then use "Mystic Box" to destroy your opponent's face-down monster and give him "Copycat" with zero attack, giving you an opening to attack his life points.
This card works well with one or multiple "Solidarity" in a Spellcaster-only deck. You can copy your opponent's monster(s)'s attack and add 800 for each "Solidarity".
Use this card with "Luminous Spark", so that "Copycat" will always be stronger by 500 ATK than the monster who is copying.
On the same note, you can copy your opponent's "Five-Headed Dragon" but choose not to attack it and instead attack their weaker monsters. Your opponent will have no choice but to attack this card with "Five-Headed Dragon" and destroy both monsters, or try to remove "Copycat" with a card effect.