Difference between two branches of $D^0$ decay

A $W^+$ boson carries a positive charge. Since $W$ bosons mediate weak interactions, in this specific case weak interactions of quarks, is well known that they induce, in the quark sector, a flavour change. This flavour change is between the up quarks $(u,c,t)$ and the down quarks $(d,s,b)$. In particular the $W^+$ changes up quarks to down quarks since the up quarks carry charge $+2/3$ and the down quarks carry a charge $-1/3$ in units of $e$. The difference between them is a positive unit of charge.

So in this specific case you can change the $c$ in the $D^0$ to a $d$ or $s$ quarks. The top quark is all another story. So wether the charm quark transforms to a down or strange changes the final result. In one case there will be a $K^-$ and in the other a $pi^-$.

Now the $W^+$ boson will become a meson (at tree level). In this case we're studying the two decays in a $K^+$ or $pi^+$.

As to why the two diagrams have a different branching ratio is encoded in the CKM matrix elements that enter the decay. In fact we have that

$$D^0rightarrow K^-,pi^+implies |A|^2propto |V_{cs}||V_{ud}| D^0rightarrow K^+,pi^-implies |A|^2propto |V_{cd}||V_{us}|$$ All the other quantities are exactly the same. Which means that

$$frac{sigma(D^0rightarrow K^-,pi^+)}{sigma(D^0rightarrow K^+,pi^-)}=frac{|V_{cs}||V_{ud}|}{|V_{cd}||V_{us}|}$$

You can easily see from the Wikipedia page that the experimental values of that CKM matrix elements are different and so the branching ratios have to be different.