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I'm currently reading The Complete Idiot's Guide to Music Composition by Michael Miller and in the "Composing With Chords" section, a reference chart for chord leading was provided and it looks like this:

These Chords       Lead to These Chords         

________________________________________

I                 | Any                 

ii                | IV,V,vii*           

iii               | ii,VI,vi            

IV                | I,iii,V,vii*        

V                 | I                   

vi                | ii,IV,V,I           

vii*              | I,iii               

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However, the book does not explain why this is the case. Searching via Google does not seem to give any answers either.

Is there a scientific or logical explanation to why these chords "lead" to the specified chords, or is it simply just a result of common patterns in popular music?

Your question can’t be answered without referring to the tension of the lead tones. The tones of a scale near the consonant intervals referred to the root have a tendency to lead to the next perfect interval. (Ti is tending to resolve into do, re to do, la to so, fi as augmented 4th toward so.) A similar tension show the steps near the 3rd and the 6th. (Fa leads to mi) The tension for a solution of a halftone step is stronger the the tension of whole tone.

That means that the altered tones have a stronger tendency to resolve into the next consonant interval (ri will lead to mi, si to la, ta to la).

These are just examples, you’ll find others by yourself.

According to Ernst Kurth (1913, 119–736) the major and minor thirds contain "latent" tendencies towards the perfect fourth and whole tone, respectively, and thus establish tonality. However, Carl Dahlhaus (1990, 44–47) contests Kurth's position, holding that this drive is in fact created through or with harmonic function, a root progression in another voice by a whole-tone or fifth, or melodically (monophonically) by the context of the scale. For example, the leading tone of alternating C chord and F minor chords is either the note E leading to F (if F is tonic), or A♭ leading to G (if C is tonic).

The leading-function of a chord and it’s tendency to resolve into another chord is a similar function as the function of the leading-tones it is consisting of.

In the link below is explained how this tension of the leading-tones as an element of chords are functioning.

The rest is up to developing as you surely are not what the title of your book says.

https://en.wikipedia.org/wiki/Leading-tone

and here is just another link explaining the doremi as relative scale that can be transferred to all scales and keys:

Movable do solfège

https://en.wikipedia.org/wiki/Solf%C3%A8ge