The way this works is that the melting 'point' for basalt is actually a sloping line in pressure-temperature space[0]. At a constant temperature, the solid-liquid phase transition[1] is such that above a certain pressure, basalt is solid and below that pressure, basalt is liquid.

At the mid-ocean ridges, some parts of the very shallow crust are ever so slightly in the solid/high pressure side of this line. With a decrease in pressure due to sea level change, the phase transition is crossed and the rock melts. This is called 'decompression melting'.

Compare this with 'decompression freezing' of water: when you open a really cold beer bottle, ice often forms in the beer. The phase transition for H2O is such that liquid is stable at higher pressures than ice (I think this relates to ice being lower density but I don't know). Opening the beer lowers the pressure, causing some water to freeze.

This same process and some similar ones is associated with a lot of volcanic eruptive phenomena. For example, landslides on volcanoes often cause explosive eruptions because the decrease in pressure causes exsolution of gases from the magma, leading to an explosive reaction. This has been hypothesized to be a big part of the St. Helens eruption, as a small eruption lead to a collapse of part of the volcano, decompressing the deeper magma chamber and leading to a much larger eruption.

On a larger timescale, it's been shown that volcanism on Iceland was suppressed during the last ice age and had a post-ice-age resurgence.

[0]: Often called the 'Clapeyron slope'.

[1]: In reality there is range of these lines bounded by the 'solidus' and 'liquidus', with a crystal mush solution akin to ice water in between.