Unusual Titration Curve Explanation
Acid (Brønsted–Lowry Acid) is a substance that gives away its $\ce{H+}$, but the environment should be able to accept that $\ce{H+}$. After it accepted it - the acid (which is now its conjugate base) can accept it back. There are 2 extremes:

The environment already has plenty of $\ce{H+}$ already. So even if acid gives its $\ce{H+}$ to the environment, it immediately receives $\ce{H+}$ back. Hence the acid always has its $\ce{H}$ and you see $\ce{HA}$ on the picture.
The environment is basic - it can take lots of $\ce{H+}$ and won't give them back to the acid. Hence acid stops being acid - it gives away $\ce{H+}$ and becomes $\ce{A-}$.

$\mathrm{pH}$ is a measure of how many $\ce{H+}$ there are in the solution. Low $\mathrm{pH}$ means there's plenty of $\ce{H+}$ (our 1st case), high $\mathrm{pH}$ means there's a &quot;shortage&quot; of $\ce{H+}$ (2nd case).
In the middle (where $\ce{H+}$ equals acid's $\mathrm{p}K_\mathrm{a}$ which in your case is 6.5) you'll have a case when 50% of $\ce{H+}$ is taken by the environment and the other 50% of acid molecules are $\ce{HA}$.
When you titrate you do exactly that - you change the $\mathrm{pH}$ of your environment which then changes how many molecules are ionized. And then at some point your indicator changes the color which means you reached some $\mathrm{pH}$ you were looking for.
Titration curves that I've seen have the volume of solvent added (your graph probably shouldn't be called Titration Curve?) - not $\mathrm{pH}$. But these are related since if you're adding a basic titrant to acidic solution you're changing its $\mathrm{pH}$.