It is a 1903 Springfield with different bolt handle. That is obviously a 1903 bolt, receiver, and barrel on the rifle. No doubt about it. It is a pretty nice job, but it looks like there is rust in the muzzle. Sorry, but there is probably no "story" behind it. Thousands were sporterized in the 1950's, many not so nice as yours.
Its age is easily determined by the serial number on the receiver.
I would keep it, especially if it was my grandfathers. He probably treasured it, but I doubt he carried it in the war.