To me 'make a point' means to say something in such a way that you accentuate what you are saying. You distinguish your 'point' from the normal conversation. So - if you said to someone 'You need to tidy your room'. You are just saying it. If you said 'Hey Kyle!' (yes?) 'I just went into your room' (yes) 'It's a real mess'. I want you to tidy it up' - you are making a point of the mess, by having a conversation about it. You can also 'make a point' of doing something. In that case, you are deliberately doing that thing, or going out of your way to do it. 'He made a point of visiting his mother every day'.

To "make a point", means that you point something out to someone. Not physically but in your speech. For example, if my family came home and put their bags and jackets everywhere, I would say, "I would like to point out to you that this house was clean when you came home". ( Meaning it was clean before you all arrived!) I am sharing information with them that I feel it is necessary for them to know. Hope this helps!