Origins
The English word 'occur' entered the language in the early sixteenth century, borrowed from Latin 'occurrere' (to run toward, to run to meet, to present itself). The Latin verb combines 'ob-' (toward, against — assimilated to 'oc-' before 'c') and 'currere' (to run), producing the literal sense of running toward something or someone.
In classical Latin, 'occurrere' had vivid physical immediacy. It meant to run to meet someone — to come running toward them, whether in greeting, in battle, or by chance. A friend could 'occur' to you on the road (run toward you, meet you by coincidence). An enemy army could 'occur' at a mountain pass (run toward you, confront you). From this concrete sense of physical encounter, Latin developed the abstract sense of events presenting themselves — happening, taking place.
English inherited primarily the abstract senses. By the sixteenth century, 'occur' meant 'to happen, to take place' (an earthquake occurred, a miracle occurred) and 'to come to mind' (an idea occurred to me, it occurred to her that). Both meanings share the underlying Latin image: something runs toward you. An event that occurs runs toward your experience — it presents itself, happens to you. A thought that occurs to you runs toward your consciousness — it arrives uninvited, presenting itself for consideration.
Development
The distinction between 'occur' and simpler synonyms like 'happen' is partly one of register (occur is more formal) and partly one of nuance. 'Occur' often implies that the event or thought arrives of its own volition — it comes to you rather than being sought or caused. Earthquakes occur. Coincidences occur. Ideas occur to people. The passive quality distinguishes 'occur' from 'do' or 'cause' — you do not 'occur' something; things occur to you.
The noun 'occurrence' (from Medieval Latin 'occurrentia') entered English in the sixteenth century and has become the standard formal word for an event or happening. 'Frequent occurrence,' 'rare occurrence,' 'strange occurrence' — the noun shares the verb's slightly formal register.
In scientific and technical usage, 'occur' has specific applications. In geology and biology, species and minerals 'occur' in particular locations — they are found there, encountered there. Gold occurs in quartz veins. A rare orchid occurs in tropical forests. This use preserves the encounter meaning: the geologist or biologist, in exploring, meets (is run toward by) the specimen.
Latin Roots
The word's relationship to its siblings in the 'currere' family is clear. Where 'occur' is running toward (ob-), 'recur' is running back (re-), 'incur' is running into (in-), 'concur' is running together (con-), and 'excursion' is running out (ex-). Each prefix redirects the fundamental act of running to produce a distinct English word. The Latin system of prefix modification is perhaps nowhere more clearly visible than in the 'currere' family.
A subtle but important point: the double 'c' in 'occur' reflects the Latin assimilation of 'ob-' to 'oc-' before 'c.' When the prefix 'ob-' precedes a root beginning with 'c,' the 'b' assimilates to 'c' for ease of pronunciation: ob- + currere = occurrere. This same assimilation produces 'occasion' (ob- + cadere, to fall: to fall toward), 'occupy' (ob- + capere, to take: to take toward), and 'offend' (ob- + fendere, to strike: to strike against). The double consonant in 'occur' is the fossilized trace of this Latin phonological process.