A short answer to the title of your question is: Asking the question 'Why?' and trying to answer it without saying 'Just because! Everyone knows!'.
In the light of this, philosophy is all about justification. In basic courses in philosophy you might hear statements like 'Basically, you can tell me the moon is made of cheese. But you will have to present good reasons to be convincive.'
This includes some steps that are normally just not taken:
- Begging the question.
- Developing an intuition towards an answer.
- Thinking of the premises implicit in these intuitions and making them explicit.
- Justifying the premises.
- Illustrating the connection between the question, the premises and the answer.
Some do this by formalization and logic, others by illustration. This is a rough sketch though, as in academic philosophy, often the first step is to attack the question itself by its premises without questioning one's own, not even taking into consideration if the question is justifiable.
Therefore it is not wrong to state that it often seems like philosophy is just a battle of statements about truth and everyone defends his 'own truth'. But this is bad philosophy. If you want to be convincing, you have to take your opponent seriously, developing a criticism out of his very own position. Alas, it is a very rare feature these times.
A good philosophical argumentation attacks the premises of every position that can be held against it (no matter how obscure it seems at a first glimpse) out of these premises by at the same time having very good justifications for its own explicit and sufficient premises. It is about knowing the enemy better than he knows himself in order to beat him. And, in addition, know youself (your implicit premises) better that anyone else.
It has to be convincing for its opponents, not dogmatic. The power of the argument should be a power of reasons, not an institutionally supported one. And in this sense it is indeed to tell people how to think things right.
Having said this, it is not by accident that authors like Aristotle, Spinoza, Hume, Kant, and Hegel are still read today, because they tried to convince by making the weaknesses of contemporary philosophical positions and their own premises explicit before by this justifying their alternative. Dewey and Sellars are examples out of the 20th century, others might add names.
The most famous example of someone who made weaknesses explicit without adding any own solution is Edmund L. Gettier. This is still better that just making and justifying statements without taking contemporary or historical challenges seriously. It is asking challenging questions.
I think your question is linked with this one, perhaps you can get more input from there, as my answer seems to fit there, too.