This is the 15289th most frequent Russian word.
поспевать
To mature, ripen, or to be on time/catch up.
поспевать means 'to keep up with something or someone' here.
Я всегда стараюсь поспевать за временем.
I always try to keep up with time.
поспевать means 'to ripen or mature' in this sentence.
Ягодам нужно время, чтобы поспевать.
Berries need time to ripen.
поспевать implies 'following or catching up with someone’s pace or flow of discourse' in this context.