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But doesn't the meaning of 'evolve' mean to 'change from one state to another'? If something changes from 'one state to another' then how does the 'original' state still exist? This is the crux of my confusion.

The answer to your conundrum lies in geographic isolation. A species will typically inhabit a large area. For lots of possible reasons a sub-group may become geographically isolated. Once this has happened each group can go its own way. For current living examples of this happening right in front of our eyes google "ring species", which is the biological term for one particular very interesting type of geographical isolation leading to speciation.