Every time a new monster or aliens or demons appears in your town / the borderlands / Garden district etc. the reason why it is here is to kill people of course. But they're killing people because it's a game too them, killing a certain number of people in a certain amount of time.
|(This is from Kamen Rider Kuuga btw)|
10 Reasons for The Game
 To prove themselves worthy to live on this new planet/dimension they are conquering.
 To advance in rank in their society.
 This is a highly ritualized form of debt collection; we owe them something from the ancient past.
 This is a highly ritualized form of mating display. Hot alien sex if they "win".
 It's a powerful demoralization tactic.
 It is a form of training from before the REAL war/invasion beings. These games are just warmups.
 They're just really sadistic and find this fun.
 This game is an intentional handicap since they're so superior to weak little humans.
 It's a ritual in and of itself- winning at the game unlocks further powers or summons more of their kind from the other world- or even quickens the return of their ruler/god.
 The beings are very mentally constricted. Certain procedures must be followed exactly. Framing a massive war or killings as a "game" allows them to process chaos better.