Rab is someone who owns. He is the owner(Al-Maalik) and that makes me a property of Allah. This is the first meaning of 'Rab'.
Second meaning of Rab is ‘murabbi’ which means someone who ensure the growth. Someone who takes care of something so it can grow. It is possible that you own something but you don’t take care of it. For e.g: Your old books are covered with dust and are kept in drawers from years. You don't even look at them even though they are yours. You are not caring them although you own them. So you have ownership but you don’t care. The word ‘Rab’ is someone who owns but also take cares of it.
Third meaning of rab is ‘Al-mun'im’ which means to give a gift. So Rab owns, take cares and also gives a gift.
Fourth meaning of rab is ‘Al-Qayyim’, the one who make sure it stays together i.e it doesn't fall apart. The one who maintain our the existence. If Allah stops taking care of us for 1 second or half a second everything will fall apart. The Heart beating from one beat to another, blood circulation in our body is all because Allah is letting it. Allah is making it. If Allah stop taking care of it for 1 second and it's all gone. So we are completely dependant on Allah constantly.
The Fifth meaning of Rab is ‘As-sayyed’, someone who has full authority. It is possible that you own something but you don't have full authority over it. You own a car but you can’t drive it however you want. You own a property but you can not build however you want. You have to follow regulations.So when someone owns something and they have full authority to do whatever they want then it can be called as ‘Rab’. And here Allah introduces himself with the word Rab.
Allah SWT is our Rab. He cares what he owns and that to continuously. And he owns everything. If our heart beats stop for a second we will die, but we are still breathing is a proof that Allah SWT continuously take cares of us and all other things that he has created. And he has created everything. Rab means someone who owns you means you own nothing. Someone who cares you while you are growing. One who gives you gift because you own nothing so everything what you have is a gift from Allah SWT. He is the one who is maintaining my and your existence( even our next breath ).
Signed
Nouman Ali Khan
Explanation of tafseer of surah fatiha