Sea floor spreading is one of the primary ways that new crust is added to the surface of the Earth. In short, it is the slow eruption of magma from the Earth's mantle along specific places in the ocean which add to the plates on either side.
Spreading is a form of volcanism, meaning that magma is erupting from the interior of the Earth through cracks or holes in the crust, but in the case of seafloor spreading there isn't a specific "volcano", and the eruptions happen very slowly. Instead of a volcano, there are rifts, which are basically long cracks in the Earth's crust. Typically these rifts are found in the middle of an ocean if they're old enough, but can be found on land if they're younger; there is a growing series of rifts in East Africa that will eventually be on the floor of a new body of water. Another well-known and documented area is the Mid-Atlantic Ridge, which is responsible for separating North and South America from Europe and Africa (and also the reason why South America fits so well with the coast of Africa when you align them)
Once a rift forms, the slow subduction of the plates on either side drags them downward due to their own weight. This pulls the rift open, leaving more room for magma to erupt, adding to the plate each time it subducts.