star - Why is the Sun's brightness and radius increasing, but not its temperature?

The effective temperature $T_mathrm{eff}$ of a star, which is presumably what's been plotted, is defined through its relationship with the star's radius $R$ and luminosity $L$ by

$$L=4pi R^2sigma T_mathrm{eff}^4$$

This comes from the assumption that the star radiates like a black body at the photosphere. While this isn't strictly true, it's quite accurate, and regardless, that's how we define the effective temperature. The actual surface temperature will be slightly different but also behave roughly as plotted.

So, even if $T_mathrm{eff}$ is constant, the star expands if it grows brighter. Also, you can see that the sensitivity to temperature is steeper than radius, so a moderate change in luminosity can be absorbed by a relatively small change in effective temperature.

While the luminosity is basically determined by the simple behaviour of the nuclear reactions in the core (in terms of temperature and density), the surface properties depend on how energy is being transported near the surface. For radiation, you have to consider what the opacity of the material is, which itself depends on ionization states and whatnot. It's easy enough to see why the luminosity grows (the core gets denser and also hotter, producing energy faster) but the determination of the surface properties is more complicated. For the Sun, it turns out the way shown in the plot after you solve all the equations with the relevant opacities.

Also, as an extreme counterexample to "brighter means hotter and smaller", remember that red giants are much brighter but also much cooler!

PS: I'm not sure the source of the data but I would guess the wiggle at the start is because of the star finishing its contraction onto the main sequence. That is, before the first minimum, energy is being released by gravitational contraction. After it, the energy from nuclear reactions starts to dominate.