Firstly, the gravitational field inside the Earth, decreases with depth.
To a first approximation, you can use the shell theorem for spherically symmetric mass distributions to argue that the gravitational field at some depth is due only to the mass enclosed within a sphere interior to that depth. If we further make the crude assumption that the Earth's density is constant, we get a simple result:
$$ g(r) = \frac{4\pi r^3 \rho G}{3r^2} = g_0\left(\frac{r}{R}\right),$$
where $\rho$ is the density, $r$ is the distance from the Earth's centre, $R$ is the radius of the Earth and $g_0$ is the surface gravity.
Although this is a crude approximation it correctly predicts that gravity eventually gets weaker towards the centre and is roughly zero at the centre. (Edit: Note that a more accurate density profile has the gravity fairly constant until you reach the core at a radius of 3500km, follows by a pseudo-linear decrease to zero at the centre).
Secondly, even though the field gets weaker, the gravitational potential is still getting deeper. Gravitational time dilation works as follows. A clock in a stronger (more negative) gravitational potential will be observed to run slower by an observer further out of the potential well, and vice versa. In this case, the observer near the core is deeper in the potential. If someone travels to the core and then comes back, their clock will have run slower compared to one at the surface.
The size of the effect is tiny for the Earth's potential well (roughly 350 pico-seconds lost per second spent at the core) but the effect is similar in size to that which is corrected for in the atomic clocks used by GPS satellites in orbit around the Earth, which are in a region of lower (less negative) potential than a clock at the surface.
