That the net force exerted by earth on the particle at that point is zero. Acceleration due to gravity is zero as well. (a = F/m)
Let me elaborate:
Suppose Earth is perfectly sphere and has uniform density. A particle is at the center of the earth. Earth is basically a collection of particles. For any particle exerting some force on the particle at the center we can find a particle symmetrically opposite to it exerting an equal force in the opposite direction to the former one. So, net force by all particles on the particle at the center is zero.
Read Gauss's law for more general theory.
Consider a particle situated anywhere and Earth.
Force at the particle (F) = mass of the particle(m) * Gravitational field at that point(E). To find the gravitational field at a point, construct the equipotential surface (call it, S) passing through that point and that would be a sphere passing through the particle with Earth's center as its center.
Now by Gauss law, Net Gravitational flux through the surface S = 4 * pie * G * (M')
where M' is the mass of the earth enclosed in the surface S and G is the universal gravitational constant.
And this flux will be equal to surface integral of the gravitational field through S. Since S is equipotential surface, magnitude of gravitational field at each point will be same, so
flux = 4 * pie * r^2 * E
So, E = G * M' / r^2. And force, F = G m M' / r^2
Net gravitational flux is always contributed by the mass inside not outside. So, at the center (M'=0), flux is zero and hence field and force as well.