It’s actually obtained by solving an eigenvalue problem. Here more info can be found on the steps involved as well as some background info on eigenvectors etc.
Usually we want to go for dimensionality reduction of the feature space to get a better ratio of data to dimensions (which can often help to mitigate overfitting when dealing with limited amount of data). We can get rid of redundant information in our features by doing this transformation to a smaller space which is spanned by the principal components (a subset of the eigenvectors of our previously mentioned problem), see also this thread: Does embedding projector use dimensional reduction? - #4 by Christian_Simonis
Here some exemplary code which you can use to play around with and e.g. check how much information is explained by which of the principal components:
(As you see, the last PCs do not provide too much value here information-wise…)
Best regards
Christian