Highly related question: Why pay a fee to have your article (in a subscription-based journal) made open access when you can just put the preprint on arXiv?
Why would I/University/professor pay for the paper to get published when it can be done without providing the fee? Why will the University pay such a sum however small it may sound to them to get my paper published when it could be done without paying for it?
This can be for a multitude of reasons. Perhaps you/the university/professor thinks, for ideological reasons, that their work should be freely available to anyone who wants to read it. Perhaps they had, in the past, tried to access someone else's paper which is very relevant to their work, but could not because of a paywall. Perhaps the funding agency that is funding the work has a policy where any publication as a result of the grant must be published open access. There is some evidence that open access articles are more widely read, and more widely read is loosely correlated with more frequently cited. And so on.
Of course you can also publish subscription. But note that subscription articles are not exactly free. They're free for you, but not for your university library. If everyone in the world published open access, there would be no subscription journals and the library would not have to pay subscription fees.
Why would I care to write to the Editor to give me a waiver to publish my article when I know that there is an option to publish it for free? Maybe I will use the money for some other purpose.
If you're using a grant for this, you don't have a choice, because you are obliged to use open access. Aside from that, if your funds are fungible, then you're free to do whatever you want with them. Just remember that if you do not publish open access, it's not freely available to readers. If you want to make your paper freely available, you have to publish open access (or ask for a waiver).
Last but not the least even if I have done a great research or path breaking work(which happens in rare cases) why would I take so much pain so that others can read my work for free? Are there really so many people in this world who care about my papers?
If people can't read your work, they won't know about whatever contributions you're making then. Your work could be lost. You don't know if there're many people who care about your papers, but you can be sure that if nobody can read your papers, the number of people who care will be zero (or maybe one, i.e. you).
There's one thing more thing I want to say, given your series of questions. Remember that open access fees are not necessarily (and in fact usually not) paid by you. If you go to your university library and ask them if they offer support for open access, they might say yes. That might make actually happens when you publish open access clearer. The library is, in principle, moving some money out of its subscription budget to pay for your open access.
