Imagine that you have two masses tied to both ends of a spring. When you pull the two masses apart and then let go, the spring force will accelerate them towards each other, right? Energy-wise, we say that the EPE stored in the stretched spring is being converted into KE.
Now imagine that you grab hold of two planets. When you pull them apart and then let go (I said imagine), the gravitational pull will accelerate them towards each other, right? Energy-wise, we say that the GPE stored in the gravitational field is being converted into KE.
For the spring, we have chosen EPE to be zero when the spring force is zero, i.e. when the spring is unstretched. As the spring is stretched, the EPE increases to become some positive value. The larger the extension, the more positive the EPE.
For the two planets, we have chosen GPE to be zero when the gravitational force is zero, i.e. when the two planets are at infinite distance apart. As they fall towards each other, GPE decrease (as it is being converted into KE) to become some negative value! The closer the two planets are, the more negative the GPE (since more GPE has been converted into KE).
The idea that GPE has a maximum value of zero (when the masses are infinitely far apart) and has negative values (when the masses come closer) may take some time to adjust to. But it comes about simply because we have chosen the maximum value of GPE to be zero. Anything else must be less and therefore negative.
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Video Explanation
xmPhysics 