RO water is water that has had all the mineral content filtered out. It should have general hardness (GH) and carbonate hardness (KH) of pretty close to zero.
When we talk about water hardness we are generally talking about GH which is how much magnesium and calcium is in the water. So RO with none will be very soft and suit some fish. Fish that prefer hard water wont suit RO water.
A different measure of hardness is KH which is carbonates and bicarbonates. KH keeps your pH stable. Things like carbon dioxide will disolve in the water, and acidify it. KH will prevent this from happening. As RO has no KH it is generally acidic and with no KH there is nothing to stop the pH dropping dramatically even further, which isnt a good thing. KH does some other things too, as its a source of carbon for a lot of the natural processes that go on in aquariums like the nitrogen cycle. So RO water with no KH stops these natural processes from happening.
Remineralising is adding salts to your RO to supply both GH and KH to water that has none. For instance chalk or limestone is calcium carbonate and will increase both GH and KH as it has both the calcium and carbonates. Baking soda is sodium bicarbonate so will only increase the KH. Epsom salt is magnesium sulphate, so will only increase the GH. There are aquarium specific products that allow you raise GH and KH independently of each other and get your RO exactly where you want with regards to pH, GH and KH.
So RO needs remineralising. You need some KH to stablise the pH, and the GH is tailored to the specific fish you are keeping.
Remineralising RO is usually reserved for reefs as tap water contains other chemicals that coral wont tolerate, and the RO process removes these chemicals. Whether you would benefit from RO in a freshwater aquarium really comes down to what your tap water is like and what fish you want to keep. Its really difficult to make hard tapwater soft, but easier to make soft tapwater hard so RO is often used where people have hard tap water, but want to keep soft water fish like discus. RO will take a lot of chemicals and contaminants out of the water, so if you have really poor quality water it might be an idea. You also shouldn't need water conditioner because the RO process will remove the chlorine/ chloramine.
You mention alkaline water, and alkaline water will usually be high GH and KH. If you want to keep fish that like harder water, like livebearers or african cichlids they will probably like your water. If you want to keep fish that prefer softer water like tetras, they would probably benefit from mixing RO with your tap water just to bring levels down a little. If you want to keep discus or other fish that want really soft water, you are then probably in the realm of needing RO with some remineralising. If its a matter of removing contaminants, again remineralising RO is a good way to go.