Bear who took in Mowgli takes another student around lair - it's got bigger (9)
I believe the answer is:
ballooned
'got bigger' is the definition.
(balloon can mean to swell or grow in size)
'bear who took in mowgli takes another student around lair' is the wordplay.
'bear who took in mowgli' becomes 'Baloo' (bear in The Jungle Book).
'takes' indicates putting letters inside.
'another student' becomes 'l' (as in L-plates for learner drivers. I am not sure about the 'another' bit.).
'around' says the letters should be written backwards.
'lair' becomes 'den' (synonyms).
'den' written backwards gives 'ned'.
'baloo' enclosing 'l' is 'balloo'.
'balloo'+'ned'='BALLOONED'
'it's' is the link.
(Other definitions for ballooned that I've seen before include "swollen up" , "piled on pounds" , "puffed out" , "swiftly developed 6D [SPARE TYRE]" .)