The present perfect is formed as has / have + V3 (past participle).
1. has learnt (or has learned)
2. has treated
3. has taught
4. has written
5. have brought
6. has become
The whole paragraph reads:
Since becoming an animal doctor, Doctor Dolittle has learnt to communicate with all animals and birds. He has treated countless farm animals and pets with care. The parrot, Polynesia, has taught him bird-language, and he has written books about his adventures. Many people have brought their sick animals to his door. Over the years, Dolittle has become renowned worldwide for his unique abilities.
Why the present perfect here: Every sentence links a past action to the present — “Since becoming an animal doctor…”, “Over the years…”. That is exactly what the present perfect is for: an action begun in the past whose result still matters now.
Watch out: Only blank 5 takes have, because its subject ‘Many people' is plural. All the others take has.