It helps me to think of it as a historical subject. We try to understand the world and ourselves. When we arrived at the scene, when the first human beings acquired the capacity to articulate thought, we tried to comprehend what was happening.
At first we told stories. Stories usually related with dietes associated with the creation of the Earth, the rivers, the sea. Gods moved the winds and sacrifices guaranteed good fortunes. Thunder was the anger of the Gods.
At some point, we gathered in large enough numbers to refine our thinking into more precise and accurate accounts of the world, not relying on myths, but on observing the world closely.
By these means, we achieved considerable success in mathematics and parts of astronomy. But many questions pertaining to us and the world remained problematic:
How can we step on the same river twice? How can we reach a target if there are an infinite number of events separating an apple from an arrow? How do we speak of one entity being the same person if they've gone crazy? How can thought arise from matter? And so on.
Fast forward thousands of years and we get science, based on observing the world under the guidance of an explanatory theory. We reduce the entities analyzed and focus on select things to study, putting aside phenomena that interfere with a theory.
We couldn't, after all, fire a canon ball around the Earth's surface. But given certain conditions (removing friction, for instance) we find out that the same force that causes an apple to fall causes the movement of the moons and planets.
But after all this, questions still remain. Important questions and hard ones. What is a self? Do we have free will? Is the world independent of me or a product of our way of ordering the world? What is a good life? How can matter produce thought? What is causation? How many things are there in the world? And so on.
Philosophy, then, is the rational enquiry into very hard question, on topics we have barely been able to make progress in for thousands of years.
The confusion on what philosophy is likely stems from the fact that some of the best philosophers in history, were also scientists. If you asked Hume, are you a scientist or a philosopher he could not say. Same with Descartes and Kant and Leibniz and Locke and many others. To them, there was no distinction.
For the Greeks even less so.
At least that's how I think of the topic.