This is a copy paste of my previous answer:

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The standard story usually told is that somewhere in the 1550s the Portuguese helped the Chinese destroy some pirates and in gratitude the Chinese granted them Macau and certain trade concessions.

Modern scholars usually either flat out reject such events even occurred, or just stress out that even if any such military anti-piracy action did happen, it was of minor importance in the normalization of relationship.

Portuguese arrival and settlement in Macau is in actuality a complicated series of events and minor intrigues in the overall socio-economic politics of china.
But it can actually be easily summarized as: portuguese bribed the local governors to allow them to stay and trade without proper royal authorization, then made themselves useful and non-threatening so there was never an official response to shut them down. With a lot of time the situation cleared up and was accepted as a de jure official status.

First contact

To go back to the beginning the first official contacts between Chinese and Portuguese happened in 1516/1517 when Portuguese sent an official embassy to China to open trade negotiations led by Tomé Pires and Fernão Peres de Andrade. Despite Malacca already have fallen, that event alone seemed not to be a deal-breaker at the time (but it was important). The Portuguese were cordially accepted, but the embassy moved slowly to actually have an audience with the Emperor, with the Emperor's absence and the apparently slow procedures of chinese court, which the Portuguese considered to be attempts by the officials to solicit bribes. During the wait, the Emperor died in 1521, and Grand Secretary, Yang T'ing-ho was appointed as interim governor. He quickly decided that only the old, traditional tributaries of the Chinese empire could reach an audience, and as Portuguese weren't it, sent them home. This decision was probably a part of court intrigues between traditional mandarins like Yang T'ing-ho who were in favor of enforcing the Ming ban on all non-royally controlled foreign trade, and the eunuchs who were much more relaxed on this issue, partly because they had their fingers in it. But I am not an expert on Ming court though, so some aspects of my simplification might be incorrect.

At around the same time as Fernão de Andrade led the embassy, his brother Simão de Andrade wasn't doing him any favors by well violently conducting his own trade, smuggling, piracy at the Chinese coast and islands, creating a really bad reputation for the portuguese. Stories circulated that the Portuguese kidnapped children to eat them alive, and while these stories were exaggerated the truth was not so much better: the portuguese were kidnapping children to sell as slaves, and while they weren't the only ones doing so, it seems the quantity and ferocity of it made them famous for it.
The death of the Emperor also brought the order to immediately stop any ongoing trade and some Portuguese ships already in China decided not to comply bringing minor conflict of these trade expeditions and the Chinese authorities in 1521, and again in 1522 when two Portuguese smaller ships were destroyed and number of cannon and men were captured. The Chinese also imprisoned the embassy members on their way back, and executed a fair number of captured men from the ships and the embassy in retaliation.

Non-presence

In the following years the Portuguese did not have an official presence in the kingdom but they joined numerous other local japanese, chinese etc. traders effected by the maritime trade ban who from various islands and local centers conducted the illegal trade and/or piracy, often with assistance of complicit chinese governors of coastal cities who profited from it. The situation remained so until the 1547 when Chinese official Chu Wan was appointed Grand Coordinator to end smuggling and piracy on the south coast. It is this when the anti-piracy action Portuguese conducted apparently happened, but there is no real record of it. Anyway, the Grand Coordinator was relatively successful in his campaign to end unsanctioned trade, which brought him many enemies who succeeded to remove him from power and imprison him in 1549. A large reversal in policy followed and trade with all foreigners resumed in the semi-official capacity (it was not officially sanctioned, but the duties and taxes were paid), with the exception of the Portuguese "Farangi" who were singled out as still being forbidden to trade.

Establishment of Macau

It is then, from 1552-1557 when actions of a newcomer Leonel de Sousa financed by Simao d'Almeida, and spurred by jesuit Francisco Xavier (before his death), made a deal (aka gave bribe) with new Chinese official Wang Po to be allowed to have trade with the Portuguese (who sort of tried to rebrand themselves as not Farangi, as well as refrain from further atrocities), first on islands of Shang-ch'uan and Lampacao , and then on territory of present day Macau. He even brokered to pay less duties, and have certain administrative rights.

The Portuguese were to send an embassy to the Emperor to get official recognition but that was stopped in Canton, when the Portuguese were asked to explain did they do what the last embassy was ordered to do, which was return possession of Malacca. Even with the failure the arrangement remained from 1550s throughout 1560s and even survived the official inspection by Chinese court in 1572 where the annual bribe was explained as a "ground rate" payment to the imperial treasury. In 1582 after ascension of the Spanish King Phillip II as king of Portugal, Chinese governor Ch'en Jui summoned representatives of Macau and reprimanded them for being on Chinese territory, however the next day, after excepting the "gifts" worth 4000 cruzados the governor "formally" praised and allowed the Portuguese to stay making another step of official acceptance.

Conclusion

The Portuguese did not "own" or "purchased" the land they just rented it and governed themselves on it. The Emperor never formally allowed it either.

Macau was to be only for harbor and disembarking the ship, while the trade and paying duties were to be done in Canton. The land they had was not to have any fortifications and was walled up with a single entrance making it entirely at the mercy of Chinese who could at any time stop the food supply and starve the city, which they did often in trade negotiations and other disputes. Chinese always had a option to conduct a military attack which would most likely succeed had they tried, but such a thing never happened. Chinese, at least the local governors and officials were comfortable with the position Macau was in, the profit it brought and the cooperation they received and let it be. The Portuguese positioned themselves as intermediaries in trade between China and Japan in silk and silver, when both nations were more skeptical of each others the the Europeans, especially after Japanese invasion of Korea in 1590s. the Portuguese used this to entrench themselves and remained and the Ming never bother to question them.

Sources:

The Cambridge History of China, Volume 8, Chapter 7: Relations with Maritime Europeans 1514-1662, page 333 and on

Foundations of Portuguese Empire, Diffie and Winius page 387-389