What is the intuition behind blowup of affine space?

Question

I learnt that the dimension of the blowup fibers “explodes” in the center of the blowup and my imagination for now goes as follows: every line on the affine plane is sent identically to a line in the blowup, except for all the lines passing through the center at which they seem to “explode”. Still, I’d like to have some more geometric intuition, if available, about what is happening here: why does this transformation provide any help for algebraic geometry? Why is it so fundamental to the nature of birational maps (to the point that, for example, every birational map can be factorized into simple blowups)? Quoting from Wikipedia page “the metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion“, which confused me even more. Any help would be very appreciated.