Conceited and shallow, needed kicking! (7-6)

I believe the answer is:

swollen-headed

I'm a little stuck... Click here to teach me more about this clue!

'conceited' is the definition.
(I know that conceited can be written as swollen-headed)

'shallow needed kicking' is the wordplay.
'shallow needed' anagrams to 'SWOLLEN-HEADED'.
But, I'm not sure how the anagram is indicated.

'and' is the link.

Can you help me to learn more?

(Other definitions for swollen-headed that I've seen before include "With an exaggerated sense of self-importance" , "Conceited" .)

I've seen this clue in The Telegraph.

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