One way to look at this problem is to consider angular momentum. The Earth spins around its axis, and has therefore some angular momentum by itself. The angular momentum is proportional to the mass of the Earth, to its squared radius, and to its angular velocity. But the Earth is not alone; it has the Moon rotating around it that adds angular momentum to the Earth-Moon system. And, even though the Moon is not as massive as the Earth (it is about 100 times less massive), nor rotates very fast around the Earth (and therefore a lesser angular velocity), it has a large orbit (about 300 000 km) and overall, it adds to the system an amount of angular momentum comparable to that of the Earth itself.
Now, think of a spinning-top: the faster it rotates (and thus the larger the angular momentum), the more stable it is. It is the same for the Earth-Moon system: without the Moon, the angular momentum of the Earth itself would be such that gravitational perturbations could be sufficient, in the long run, to significantly perturb its axis (exactly like the spinning-top: if it does not rotate very fast, a small perturbation will rapidly increase and the spinning-top axis will start to oscillate more and more). But with the Moon, the global angular momentum of the system is larger, and it is therefore harder to sufficiently perturb the system to get it to oscillate strongly.
For those who wants the dirty details, you can have a look at Laskar et al. 1993.