That image is based on a common one shown in Omani schools, but it's honestly a little optimistic. The real, effective level of control over parts of the Persian coast, and especially Africa are questionable. Here's a post on Omani history I wrote a little bit ago as a quick foundation. It gives a brief overview of how the Omanis managed to become a regional power after kicking out the Portuguese.

Regarding the province of Hormuz, it was attacked and ultimately taken over by the Portuguese, under the command of Alfonso De Albuquerque as a result of its mercantile policies aimed at stopping Portuguese expansion in Asia, as well as its inherent wealth. Ismai'l I Safavid was busy in a war with the Ottomans and unable to retake it, though it did manage to rebel and remain independent for a few years. The Portuguese retook it, replaced the native Persian leadership with loyal Omanis, and held it until Shah Abbas I captured it with the help of the English in 1622.

In the late 1700s, during the war to push out the Persians, the Omanis retook parts of the Persian coast, and managed to hold it until the mid to late 1800s, when the Persians managed to retake it. I'm sketchy on the precise years and details unfortunately, as it's usually a footnote in more mainstream history books if it's mentioned at all.

Sources are the same as in the earlier post: The New Cambridge History of Islam and History of the imams and seyyids of Oman: A.D. 661-1856 by Salil ibn-Raziq and translated by Hamid ibn Muhammad ibn Ruzayq