What "the inherent dimensionality of data points will be smaller than the number of coordinates" means is this: If you have 2 dimensional data, you need a 2 dimensional coordinate system (x,y for example) in order to be able to show them. if you data has n dimensions, you need an n dimensional coordinate system, where after n=3 it is impossible for us to visualize that geometrically (at least for me). So if you have high dimensional data, it is diffucult to show it. However, there is a trick to that. What you can do is to say: OK, I might need a lot of dimensions to show the data in the usual corrdinate system, but is there a way to transform the data into some other form, so I will need less axis to show it, or similarly an alternative coordinate system, where I can show my data with less axis? The answer is yes, and PCA is how you do it. You say, OK, let me find the direction where the change in my data is the highest, i.e. the varience is highest, and use this direction as the first axis of my new coordinate system. Theen you say, OK, now I have a direction that explains some (most) of the variance in my data, but what in which direction is the variance of the data second highest, you find it and it is the second direction of your coordinate system. Then you repeat it several times and come up with several axes that explain the most of the variance of your data. The rest of the varience in your data is very small and not so relevant, as you already have most of the change in your data. So now, in this new system, you have nearly all the variance of your original data, however you have less axes, i.e. your data is nearly as well represented in this new corrdinate system as in the previous, however it has significantly less axes in the new one. This is what is meant by your data having inherently less dimensions (in the new coord. system) than the number of coordinates (in the old coord. system).
This representation in the lower dimensional new coordinate system is allowed by the fact that your data actually has some dependencies in the previous corrd. systems bases, i.e. your data can be more efficiently represented in another coord. system with other basis functions. PCA is a special way of finding these basis functions that define the new coordinate system.