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I read so many articles and the Fast R-CNN paper, but I'm still confused about how the region proposal method works in Fast R-CNN.

As you can see in the image below, they say they used a proposal method, but it is not specified how it works.

What confuses me is, for example, in the VGGnet, the output of the last convolution layer is feature maps of shape 14x14x512, but what is the used algorithm to propose the regions and how does it propose them from the feature maps?

enter image description here

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Yes, it is not specified because the region proposal algorithm did not change from R-CNN (the previous version from Fast R-CNN, however, in the next verion, Faster R-CNN, this algorithm is replaced by a CNN).

The region proposal algorithm you are looking for is called selective search. You can find in the R-CNN paper that the algorithm is described in "Selective Search for Object Recognition", I found a copy here.

The algorithm is based on a series of segmentation and aggregation techniques of the input image for generating the proposed regions. Check it out 4 iterations of segmentation & aggregation over the same input image to build the proposed regions.

enter image description here

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