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"he's had a few" is an idiomatic expression meaning "he has had a few alcoholic drinks", so it means that this person is a bit drunk now.
The present perfect tense [現在完了形] is used because it's a completed action which caused a direct effect in the present.
Compare:
He's had a few drinks. (→ he is a bit drunk now)
Yesterday, he had a few drinks. (→ he is probably sober now)
❌ Yesterday, he's had a few drinks. (→ WRONG English)
He has gone crazy. (→ he is crazy now)
He went crazy yesterday. (→ he became very angry yesterday (but maybe he's calm now))
