Distance between quotation mark and text

The distance between the quote and the following letter depends on the kerning information of the font. This is a decision (or missing decision) of the font designer. csquotes also provides a fallback setting, if the font does not have a corresponding kerning pair. The fallback can be set by the optional argument of between the closing outer mark and the opening inner mark of \DeclareQuoteStyle. The default definition of (the American variant of) style english at csquotes.def is

\DeclareQuoteStyle[american]{english}% verified
  {\textquotedblleft}
  {\textquotedblright}
  [0.05em]
  {\textquoteleft}
  {\textquoteright}

Changing the 0.05em does not change anything in your example. So it seems, that for European Computer Modern (the font you are using because of \usepackage[T1]{fontenc} the optional fallback kerning is not used.

Nevertheless you could force another kerning after \textquotedblleft using the \kern primitive:

\documentclass[11pt,openright]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[a4paper]{geometry}
\geometry{verbose,tmargin=3cm,bmargin=3.5cm,lmargin=4cm,rmargin=3cm,marginparwidth=70pt}
\usepackage[autopunct=true]{csquotes}
\DeclareQuoteStyle[american]{english}% verified
  {\textquotedblleft\kern0.01em}% force additional kerning
  {\textquotedblright}
  [0.05em]% Fallback only for fonts without kerning
  {\textquoteleft}
  {\textquoteright}

\begin{document}
    \raggedright
    Some text \textquote{text of quotation}.\\ Some other text \enquote{text of quotation}.\\ Some text \textquote{other text of quotation}.\\ Some other text \enquote{other text of quotation}.
\end{document}

But note: This would add extra kerning independent of the font and independent of the letter that follows the double quote!

An alternative would be to use a font with improved kerning, e.g., Latin Modern:

\documentclass[11pt,openright]{book}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[a4paper]{geometry}
\geometry{verbose,tmargin=3cm,bmargin=3.5cm,lmargin=4cm,rmargin=3cm,marginparwidth=70pt}
\usepackage[autopunct=true]{csquotes}

\begin{document}
    \raggedright
    Some text \textquote{text of quotation}.\\ Some other text \enquote{text of quotation}.\\ Some text \textquote{other text of quotation}.\\ Some other text \enquote{other text of quotation}.
\end{document}