The key bit of the Evolution is the last part: "Discard this Evolution on anyone's turn to force a Monster to flee Tokyo when damaged."
It can only be played when a Monster is damaged. The most likely way a Monster gets damaged is through being attacked. And being attacked triggers the "stay" or "leave" option, except now you can force them to choose "leave", making the attacker take their place.
I will admit that as worded the card can work on non-attack-based damage effects, but it was most likely designed with the intent it only works on attacks. You'll have to houserule what to do.
Going by the (slightly) revised 2nd Edition rules, it depends on when the damage occurs. If the damage happens during Step 2 "Resolve Dice", then the attacker would have to enter in Step 3 "Enter Tokyo", as Tokyo would be empty. If the damage happens during step 4 "Buy Cards", then Tokyo would remain empty until the next player's turn, where they would be required to enter Tokyo.