Why are there only a few known Ramsey numbers?

Unfortunately, the original proof that Ramsey numbers exist (as in, are finite) was non-constructive, so mathematicians have been fighting an uphill battle from the beginning. Consider the diagonal Ramsey numbers $R(s, s)$. The smallest of these which remains unknown is $R(5, 5)$, so let's say we suspect $R(5, 5) = 43$—the current lower bound. We would need to confirm that the relevant property holds for every possible $2$-coloring on $K_{43}$. Since this graph has $\displaystyle \binom{43}{2} = 903$ edges, there are $2^{903}$ possible colorings. This is a $272$ digit number! For comparison, this number is many orders of magnitude larger than the total number of protons, neutrons, and electrons in the entire observable universe. Of course, it's possible we can whittle this number down a bit with some ingenuity (for instance, some colorings are equivalent up to symmetry), but it would still be unimaginably gargantuan. Brute force is out of the question, even for the most powerful supercomputers.

Bounds:

The work on lower bounds for diagonal Ramsey numbers improves upon the original lower bound Paul Erdős found using his probabilistic method$^\dagger$. With this method, it's possible to show (rather easily) that $\displaystyle R(s, s) \geq \lfloor 2^{\frac{s}{2}} \rfloor$. Of course, the general lower bound has since been improved, but it still has an exponential growth factor of $\sqrt{2}$.

A relatively decent upper bound for diagonal Ramsey numbers can be proven using the same approach as in the proof that $R(s, t) < \infty$. That is, we can show $\displaystyle R(s, t) \leq \binom{s+t-2}{s-1}$. When $s=t$, we get $\displaystyle R(s, s) \leq \binom{2s-2}{s-1}$, which grows exponentially with a growth factor of $4$. The current upper bound has been improved a bit, but still has the same exponential growth factor.

$^\dagger$ I'd highly recommend checking out this proof; it's one of my favorites. A downright ingenious application of probability theory to this area of math. Find it here.