French present of indicative e.g., je parle, tu parle, etc., may be interpreted as
I am speaking, I do speak, I speak according to the context.
The use of bien here is to give a clearer emphatic tone.
J’ai bien peur de devoir partir maintenant.
I’m afraid that I have to go indeed.
As another example:
Je lui ai bien dit de s’arrêter.
I did tell him to stop/I told him to stop indeed.
The adverb bien can have either an emphatic or a softening role depending on the context.
In that particular case, that’s the second possibility so j’ai bien peur is weaker than j’ai peur, meaning you are not really afraid of anything but simply state there is probably no alternative and you just have to leave now.
Bien plays a similar role when used with aimer:
J’aime l’Italie – I love Italy.
J’aime bien l’Italie – I like Italy.