There is no simple answer to this question. Below is a brief survey of three different types of philosopher. Each has different motivations for their work and so do their followers, which I have speculated about below.
The political philosopher
Philosophy and politics (and economics) are interwined; philosophy is intensely practical in a socio-political sense.
Prime examples: Karl Marx, the later Jean-Paul Sartre.
Other examples: Plato (The Republic), Aristotle (The Politics), Adam Smith, Michel Foucault, Noam Chomsky
Motivations: Anger, digust and resentment in regards to the treatment of certain people within society, founded on notions of egalitarianism and compassion; despair at what society has become; desire for order, security and efficiency (excellence); visions of a better world.
Philosophy as a guide to living
Philosophy is supposed to do something for you, as an existing individual. It is much ado about ethics. Philosophy is virtually inseperable from psychology.
Prime example: Existentialists (Kierkegaard, Nietzsche, Heidegger, Sartre, etc.)
Other examples: Socrates, Aristotle, Stoics, Epicureans, Kant, Schopenhauer, Freud.
Motivations: egoism; disgust at the fraudulence and negative outcomes of ‘herd behaviour’; disappointment that humans haven’t reached their potential and are only living half-lives; disgust at the victimhood mentality, strong ethical sense of responsibility; desire for the maximisation of personal meaning and pleasure; self-help and personal development; consolation; the search for a remedy to the distress of existence.
Philosophy is the analysis of concepts
Academic, professional philosophy, chiefly concerned with logic and mathematics, but also linguistics and epistemology.
Prime example: Bertrand Russell
Other examples: the majority of profressional philosophers at universities today
Motivations: shelter from the complexity, stress and social pressures of the real world; cosy university position; prestige amongst peers; pride in self-perceptions of being smarter than others; disdain for less intellectual types; the dictates of industry.