You can use the in operator.
I took the liberty of also adding case-insensitivity by lowercasing both the game name and the user input.
list_of_games = [
    "Super Mario Brothers",
    "Animal Crossing",
    "Legend of Zelda Breath of the Wild",
    "Kirby Superstar Ultra",
]
search_string = input("Search for a game:").lower()
for title in list_of_games:
    if search_string in title.lower():
        print(title)
As discussed in the comments, if you'd like to handle things differently depending on how many games match the input, we can change things e.g. so:
search_string = input("Search for a game:").lower()
# Build up a list of matching games using a list comprehension
matching_games = [title for title in list_of_games if search_string in title.lower()]
if not matching_games:  # the list is falsy if it's empty
    print("No matches for that input, sorry!")
elif len(matching_games) == 1:  # Only one match
    print("One match:", matching_games[0])
else:
    print("Multiple matches:")
    for title in matching_games:
        print("*", title)