It's the conditional - "he would be a doctor if (condition=true)".
The conditional is used to express a preference in expressions like ba mhaith liom cupán tae - "I would like a cup of tea", or b'fhearr liom caife - "I would prefer coffee"
ba mhaith leis bheith ina dhochtúir - "he would like to be a doctor"
I think knowing how the hover notes are constructed — apparently by blind dictionary lookup rather than with any reference to the larger context of the example being presented — might be a matter of some interest to learners. This certainly puzzled (and sometimes annoyed) me when I first began working through the lessons. Learning to deal with this sort of distraction is part of the game, eh?
Is dochtúir é - "he is a doctor" Present Tense
Is é an dochtúir é - "he is the doctor" Present Tense
(in Ulster Irish, they just say Is é an dochtúir).
Ba dhochtúir é - "he was a doctor" Past Tense
Ba é an dochtúir é - "he was the doctor" Past Tense
(in Ulster Irish, they just say Ba é an dochtúir).