I think maybe I'll take the opportunity to give my intuitive derivation (very non-rigorous and objectionable, but cute), which I don't think you will see anywhere else (at least not in this form), since I came up with it. To save time, I'll be fairly sloppy.
I'll just do the case of one particle confined to a move in a straight line. You want to choose a path q(t) that is extremal for the functional
\int L(q,\dot{q},t) dt
Now, you want to perturb the path by a little bit and the integral doesn't change (to a first order approximation). You can imagine that the path is formed by a bunch of vectors, dq_i, going tail to tip, each one being traversed in time dt. So, imagine you make one of the vectors longer, starting at time, t, so that q gets bigger (make the next vector have the same tip). This makes the integral go up by \frac{\partial L}{\partial q}.
But, you have also changed \dot{q} because the velocity vector is longer. So you add \frac{\partial L}{\partial \dot{q}} at time t and you subtract it at time t + dt.
So you have three terms that are changing the integral, but you can see that the latter two are just -\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} when dt is small.
So, there you have it: \frac{\partial L}{\partial q}-\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = 0
Quite a beautiful and transparent (perhaps, after some thought) argument, I find, if you are well-prepared enough for it. The case of more particles and dimensions is not significantly different.
Ah, the wonders of not being rigidly formal all the time...
