Use "Mystic Box" to give your opponent one of your monsters, then Special Summon another of your monsters with an equal Level from your hand; this works particularly well when the monster in your hand is stronger or if you have another high level Monster in your hand that you can tribute it for.
To be able to summon a certain monster you could give your opponent a monster of the same level first. Some possibilities are already described above. Other ways would be i.e. "Creature Swap", "Ojama Trio" or "Revival Gift".
Use this card to summon monsters to the field that need a companion on the field like i.e. "Boot-Up Soldier - Dread Dynamo" that needs a gadget or like union monsters. For Synchro summons or XYZ summons you also need at least two monsters on the field.
Use this card to summon a monster to the field at the time its effect can activate like i.e. "Spirit of the Breeze" in attack mode during your standby phase or a "Mirage Dragon" after entering your battle phase.