Here's a quick and dirty version of George Lowther's calculation that I learned from Bjorn Poonen. It is presented in a bit more generality in the next-to-last slide of this talk, so it is in a sense pre-masticated for colloquium use:
Let $B$ be a large positive integer which we will allow to grow. The set of integer points in the box $[-B,B]^{\times 3}$ has size of order $B^3$. A homogeneous degree $d$ polynomial in 3 variables with integer coefficients (describing a degree $d$ plane curve over $\mathbb{Q}$) will take values of size about $B^d$ when inputs are taken from the box. If we assume the values are uniformly distributed, the expected number of zeroes in the box is $B^{3-d}$.
There is a qualitative difference in expected number of zeroes based on the sign of the exponent $3-d$. This suggests that plane curves of degree more than 3 (namely those of genus at least 2) will only have "accidental" rational points.
