This is an example of what can be called associative narration, that is common in the texts of continental tradition in philosophy to which Camus belongs. Its aim is to foster understanding rather than to pass information, to invoke what one has in mind in the reader, and to help them explore it for themselves. The text is often structured as a search for words that make the right impact, or impacts from different perspectives, where explicit descriptions may be ineffective or even counterproductive. Generally, the discipline that studies interpreting of philosophical and literary texts is called hermeneutics.
"Vanity of the word experience" is not literal of course, a word can not be literally vain. But what vanity represents (in humans) is people's overconfidence in their ability to perform or create something. What Camus wants to convey is that experience proper is gained not by performing (an experiment), or creating (say a piece of art), but in fact by the opposite, not by effort but by non-effort, reception, suffering, "patience rather than experience". "We wait patiently" means to convey passivity, but grammatically uses active voice. So Camus reduces the tension by switching to "we are patients", where "are" is not even grammatically a verb of action, just a copula, even at the expense of conferring a somewhat unconventional meaning on "patient" (Heidegger might have bent the dictionary even further, into an actual passive voice, something like "we are being patiened").
The next sentence, "when we emerge from experience we are not wise but skillful", hints at a related feature of experience by associating to the distinction between wisdom and skill. Wisdom is a good judgement about what to do, while skill is the habitually good doing of it, it is skill and not wisdom that emerges from experience, or perhaps that is being experienced, according to Camus. This is somewhat related to the distinction between knowledge-that (something is the case) and Ryle's knowledge-how (to do something), or Polanyi's tacit knowledge, although wisdom is not quite pure knowledge-that, see What is the relation between 'knowledge-that' and 'knowledge-how'?
A word of caution. While associative narration is often effective for evocative and expository purposes, it does not replace argumentative reasoning about what is being evoked. When taken for that it turns into "reasoning by loose association", where incidental similarities are mistaken for revealing or representing something "deep". Nirenbergs call it the Pythagoric snare, after the loose associations of various things with numbers attributed to Pythagorean numerology.