Language emerged suddenly, as a rabbit out of a hat. This would be possible because humans are endowed with an innate language faculty (Chomsky).
Language is the result of a long process, creeping upwards in complexity over millennia, like a snail creeping up a wall. Language capacity increased very slowly.
The most accredited theory is that of a ‘language bonfire’. After this theory, sparks of language have been flickering for a long time before language begun a fast evolution and then stabilized around 50,000 years ago.
Language evolution alternated between periods of stasis (stagnation) and periods of fast development (Eldredge and Gould).