It isn't, I'd suggest, quite the case that 'Socrates started to study music (poetry/composing) just on the direction of a voice that he heard in his dream.'
A key point is that he begins writing lyrics as his latest response to a recurrent dream (60E).
Socrates explains that he has been composing
lyrics because of a recurring dream in which the
dream itself speaks to him and says: "Socrates,
practice and cultivate the arts." This is not a new
dream triggered by the harrowing events
surrounding his trial and sentence but one that has occurred "often" in the course of his life. Socrates
has clearly spent a great deal of time considering
the meaning of this dream but until now he has
thought that the "arts" referred to philosophy. Now,
though, he is not so sure, so he begins writing lyrics
in order to "discover the meaning" of this dream and
to "clear" his "conscience." Socrates takes the dream
seriously because he states that he is "obeying" it, and his conduct bears this out. (Sheldon Nahmod, 'The Dream Motif in "Phaedo"', Classics Ireland, Vol. 1 (1994), pp. 74-89; 75-6.)
It is not clear, to me at least, why Socrates takes the recurrent dream so seriously but in taking up the writing of lyrics, a new departure for him, he is trying afresh to interpret the dream. He previously thought that the 'arts' referred to philosophy. Now, for a reason that isn't evident, he is uncertain that philosophy was meant - at all or at least exclusively. In an experiment he starts on his lyrics in order to explore a different possible reading of the dream.
It looks irrational to take dreams, even a recurrent dream, so much to heart. But rationality is contextual. Rationality is never free-floating but is always informed by beliefs we have, by what we have assimilated from the intellectual or cognitive community in which we have been educated. We can work our way out of such beliefs (or some of them) but only slowly and partly.
All of which is a lead-up to saying that the Greeks of Socrates' time had a deep belief in the divine or prophetic nature of dreams. There was even an activity of mantike, that of the skilful and insightful interpretation of dreams. In Homer, before Socrates' time but an abiding cultural influence, Zeus was the sender of dreams - even if sometimes of lying dreams as in the dream sent to Agamemnon (Iliad, ii.5 ff.). Prophetic dreams make an appearance in Plato's Republic, IX.572A ff.
Socrates' 'irrationality' diminishes if we contextualise him intellectually and culturally. So at least I suggest.