From the biblical start of humanity people have been doing things for the sake of love. Kingdoms and aspirations have came and gone due to love, and video game characters are no different. In real life if a gang of thugs punched your lady love in the gut and hoisted her over their shoulders you wouldn't go chasing after them? Most would, despite the fact that it would be smarter to alert the authorities... then again where's the fun in that when you're leaving a trail of carnage in your wake on the way to reclaim your loved one.
Thinking about it, it's mostly a male dominated thing; Boy meets girl - girl gets kidnapped by by thugs, or demons that interrupt your flower picking sessions, or giant turtle/dragon things on a year to year basis. Boy goes on to fight his way to save girl. Tear-jerking scene of reunion commences. many mediums have followed this formula, and yet for an antiquated theme it seems to work, because of course it's a natural instinct to want to protect the ones you care about. which seems to make it even more compelling. Despite it's male dominated nature there are instances of females running to the aide of their beaus as well. Super Princess Peach was the most notable, with the titular character chasing after her longtime rescuer, Yuna was trying to find a way to bring Tidus back to her in Final Fantasy X-2 (it's also interesting to note that both those games bucked trends and also were very successful ventures selling into the millions).
So think about all those times you're running around the mushroom kingdom, or jumping barrels chasing gorillas through construction zones, or beating up gangs of rejects from The Road Warrior, just remember that you're doing it for love.
Happy Valentines day!