What does "heights" mean here? Should I always put "s" to use like this meaning?
Yes, in this type of context, "heights" should always appear in the plural form. The speaker has used "heights" to refer to the state of being high up, like in the sky or on a high building. This is why the full expression is "fear of heights." Ex: I have a fear of heights. Ex: I'm not scared of heights. I've even gone bungee jumping and sky diving.