Intuition behind the PCA algorithm
In simple words, PCA (Principal Component Analysis) is a dimension-reduction method where you try to reduce the number of variables from a larger number to a smaller one without any loss of the important information available in the data. 
Mathematically, it is a projection of a higher-dimensional object in a lower-dimensional vector space.

A great example in a real-life scenario is where we see 3D objects in a 2D Television. Reference

Another good explanation on the intuition on PCA 

Hope it helps.