Why does capillary action occur better in narrow tubes than in wide tubes?
You have to look which forces act where. You have a force that tries pull the liquid up the capillary, which stems from the gain in energy due to adhesion. At the same time, however, gravity acts against this force. In equilibrium, both forces have the same value:

$$F_\text{g} = F_{\text{surf}}$$
$$ mg=\rho Vg= \rho R^2\pi h g=2\pi R\sigma_s \cos(\varphi)$$

Which leads to the formula presented by Babounet in another answer:

$$ h = \frac{2\sigma_s\cos(\varphi)}{R\rho g}$$

So, to boil it down, if you lower the radius $R$, you lower the force due to gravity $F_g$, but the force due to surface energy $F_\text{surf}$ does not fall as fast, which leads to a higher height $h$.


(source, $W=F_g$ here, naming isn't perfect, $F_{\text{surf}}$ is not drawn into)