I belive that is has to do with Hooke's law F = -kx

which is the force exerted by the spring that is streched past its equilibrium position.
and the negative sign in the equation tells us that the force is a restoring force, that is the force is always directed back toward its equilibrium position.

so if you have a spring at position x= 0, and you add a mass m to that, you are going to strech it beyond its equilibrium position. so according to Hookes equation the spring is going to want to return to it's equilibrium positon of x=0, so it will accelerate back toward its equilibrium position, which is also known as simple harmonic motion or oscillation.

and if rewrite hooke's law you can see what it's acceleration is

F = -kx
F = ma
ma = -kx
a = -(k/m)x