If I had to choose a definition from a dictionary, it'd be this:
> ****
But it is far from enough specific for your question, I guess.
The explanation you got "to introduce the aim of an endeavor" is basically correct, that the expression implies that someone is _doing_ something at the moment, and tells what condition they are in (by adjective), trying to achieve or finish what (by ).
> _The teacher is busy marking the tests._
> _The manager is eager in improving scoring ability._
The adjective has to describe a certain manner of an action, so "action-less" kind of words like mental adjectives are not allowed.
> ×
>
> "I guess an izakaya in an outlying area must be desperate for differentiating itself from others."
> "I guess an izakaya in an outlying area must try the hard way to differentiate itself from others."
> "I guess an izakaya in an outlying area always struggles to differentiate itself from others."