A "contradictio in adjecto" (literally, a 'contradiction of adjectives') is simply what is commonly termed a "contradiction in terms". So, for example, a married bachelor, or a runny solid (I know there are probably ways in which these phrases could make sense, but the point is that some things are conceptually impossible).
Now, Nietzsche wants to use this idea to point out that certain philosophers are guilty of supposing the impossible in order to justify their ideas. So, Descartes for instance argued that "I think therefore I am" (the Cogito) was a case of "absolute knowledge" and "immediate certainty". It also implies that the self (the "I") is the uncaused cause of thought - a causa sui. These three examples are, for Nietzsche, contradictions in terms: you cannot have absolute knowledge (only knowledge relative to your perspective - 'perspectivism'); you cannot have immediate certainty (because knowledge always relies on interpretation, assumption, and deduction); and you cannot have a causa sui (because the notion of cause implies effect, and vice versa).
Nietzsche sees these examples as springing from a sort of 'never never land' of the intellect. In other words, they are made up to try to justify a set of personal prejudices and desires (a desire for truth, certainty, etc.). But, he asks, even if these things could be proven, the more interesting thing is why you should choose to do so! It's like psychologists who look for mathematical ways of understanding human behaviour: even if you could do this, why would you want to? Because of deeper desires or motives (probably unconscious). The same goes for philosophy.
For Nietzsche, we must avoid what he calls 'reification' - the tendency to treat ideas as if they were things. Quantum physics understands this danger: many of the ways in which we think of subatomic particles has no analogy to the way we see everyday reality - the concepts are just ways of speaking. This is hard to grasp, but Nietzsche's point is that we should not mistake words and ideas for things.
Now, you then ask a very good question: does this mean that, because words are not things, that things exist separately from words, and - perhaps - as Kant argued, we can never experience the 'real' world directly. Well, Nietzsche resists this temptation. Obviously, in a trivial sense, words are not things, but in a philosophical sense, we should not create a metaphysical world where the 'thing in itself' resides, free from all human interpretation. This is a discussion beset with paradoxes and traps: if we claim that there is no such thing as the real world, then it looks like we are left with solipsism (only I and my experiences are real); if we propose that some independent world exists, then we run the risk of metaphysical assumption (how can we know that, or even what it is like?). Nietzsche seems to want to close the gap between perception and reality (perspectivism), and to therefore avoid metaphysics. However, he also (presumably) wants to avoid solipsism and idealism (the idea that only ideas are real, or that everything that exists can be experienced). It seems difficult to do both, but you can see where he's coming from: he simply wants to avoid philosophical traps and delusions. I think for instance, with freewill, he simply wants to say, 'Look, we created these ideas to help us, not to confound us, so let's not get caught up on definitions.'
Sorry for the length - complicated topic...
