How to add additional space between the lines inside the matrix?

I propose this variant, always based on pmatrix and pmatrix* environments, but also on cellspace, which enables you to define a minimal vertical spacing at the top and bottom of cells in columns with specifier prefixed with the letter S (or C if you you load siunitx). The [math] option extends this to the various matrix environments, but not to the matrix* environment. However a small patch (communicated by the package author) makes it work also in the latter case.

Another improvement uses \mfrac (medium-size fractions) from nccmath, to avoid the size discrepancy between fractions and integers in arrays and matrices.

\documentclass{article}
\usepackage{mathtools, nccmath} 
\usepackage[math]{cellspace} 
\setlength{\cellspacetoplimit}{3pt}
\setlength{\cellspacebottomlimit}{3pt}
\makeatletter
\edef\@tempa{%
      \catcode`:=\the\catcode`:\relax
      \catcode`_=\the\catcode`_\relax}
\catcode`:=11
\catcode`_=11
\def\MT_matrix_begin:N #1{%
      \hskip -\arraycolsep
      \MH_let:NwN \@ifnextchar \MH_nospace_ifnextchar:Nnn
      \array{*\c@MaxMatrixCols {>{$}S#1<{$}}}}
\@tempa
\makeatother

\begin{document}

\begin{equation}\label{eq25}
\begin{aligned}
& \begin{pmatrix*}[r]
3 & -3 & 0 & 1 \\
\frac{16}{9} & -\frac{20}{9} & 1 & 0 \\
-2 & \frac{1}{2} & 0 & 0 \\
2 & -1 & 0 & 0
\end{pmatrix*} \begin{pmatrix*}
y_{n+1} \\
y_{n+2} \\
y_{n+\frac{8}{3}} \\
y_{n+3}
\end{pmatrix*}
= \begin{pmatrix*}[r]
0 & 0 & 0 & 1 \\
0 & 0 & 0 & \frac{5}{9} \\
0 & 0 & 0 & -\frac{3}{2} \\
0 & 0 & 0 & 1
\end{pmatrix*} \begin{pmatrix*}
y_{n-3} \\
y_{n-2} \\
y_{n-1} \\
y_n
\end{pmatrix*} \\ \nonumber
&+
h^3 \begin{pmatrix*}[r]
\frac{47}{100} & \frac{23}{40} & -\frac{81}{800} & \frac{1}{20} \\
\frac{7}{27} & \frac{203}{729} & -\frac{25}{324} & \frac{65}{2187} \\
\frac{1399}{4200} & -\frac{23}{168} & \frac{783}{5600} & -\frac{17}{2800} \\ -\frac{109}{120} & \frac{61}{120} & -\frac{81}{160} & \frac{79}{360}
\end{pmatrix*} \begin{pmatrix}
f_{n+1} \\
f_{n+2} \\
f_{n+\frac{8}{3}} \\
f_{n+3}
\end{pmatrix} + h^3 \begin{pmatrix*}[r]
0 & 0 & 0 & \frac{1}{160} \\
0 & 0 & 0 & \frac{31}{8748} \\
0 & 0 & 0 & -\frac{13}{224} \\
0 & 0 & 0 & -\frac{451}{1440}
\end{pmatrix*} \begin{pmatrix}
f_{n-3} \\
f_{n-2} \\
f_{n-1} \\
f_n
\end{pmatrix} \nonumber %\eqno{(25)}
\end{aligned}
\end{equation}
\bigskip

\begin{equation}\label{eq26}
\begin{aligned}
& \begin{pmatrix*}[r]
3 & -3 & 0 & 1 \\
\mfrac{16}{9} & -\mfrac{20}{9} & 1 & 0 \\
-2 & \mfrac{1}{2} & 0 & 0 \\
2 & -1 & 0 & 0
\end{pmatrix*} \begin{pmatrix*}
y_{n+1} \\
y_{n+2} \\
y_{n+\frac{8}{3}} \\
y_{n+3}
\end{pmatrix*}
= \begin{pmatrix*}[r]
0 & 0 & 0 & 1 \\
0 & 0 & 0 & \mfrac{5}{9} \\
0 & 0 & 0 & -\mfrac{3}{2} \\
0 & 0 & 0 & 1
\end{pmatrix*} \begin{pmatrix*}
y_{n-3} \\
y_{n-2} \\
y_{n-1} \\
y_n
\end{pmatrix*} \\ \nonumber
&+
h^3 \begin{pmatrix*}[r]
\mfrac{47}{100} & \mfrac{23}{40} & -\mfrac{81}{800} & \mfrac{1}{20} \\
\mfrac{7}{27} & \mfrac{203}{729} & -\mfrac{25}{324} & \mfrac{65}{2187} \\
\mfrac{1399}{4200} & -\mfrac{23}{168} & \mfrac{783}{5600} & -\mfrac{17}{2800} \\ -\mfrac{109}{120} & \mfrac{61}{120} & -\mfrac{81}{160} & \mfrac{79}{360}
\end{pmatrix*} \begin{pmatrix}
f_{n+1} \\
f_{n+2} \\
f_{n+\frac{8}{3}} \\
f_{n+3}
\end{pmatrix} + h^3 \begin{pmatrix*}[r]
0 & 0 & 0 & \mfrac{1}{160} \\
0 & 0 & 0 & \mfrac{31}{8748} \\
0 & 0 & 0 & -\mfrac{13}{224} \\
0 & 0 & 0 & -\mfrac{451}{1440}
\end{pmatrix*} \begin{pmatrix}
f_{n-3} \\
f_{n-2} \\
f_{n-1} \\
f_n
\end{pmatrix} \nonumber %\eqno{(25)}
\end{aligned}
\end{equation}

\end{document} 

With considering the mathtools package you equation can be written as follows:

\documentclass{article}
\usepackage{geometry}           % <---
\usepackage{mathtools}          % <---

\begin{document}
\begingroup                     % <---
\renewcommand\arraystretch{1.3} % <---
    \begin{multline*}           % <---
\begin{pmatrix*}{r}             % <---       
3               & -3            & 0 & 1 \\
\frac{16}{9}    & -\frac{20}{9} & 1 & 0 \\
-2              &  \frac{1}{2}  & 0 & 0 \\
2               & -1            & 0 & 0
\end{pmatrix*}\begin{pmatrix}
                y_{n+1}             \\
                y_{n+2}             \\
                y_{n+\frac{8}{3}}   \\
                y_{n+3}
              \end{pmatrix} = \begin{pmatrix*}{r}
                                0 & 0 & 0 & 1               \\
                                0 & 0 & 0 &  \frac{5}{9}    \\
                                0 & 0 & 0 & -\frac{3}{2}    \\
                                0 & 0 & 0 & 1
                               \end{pmatrix*}\begin{pmatrix}
                                                y_{n-3} \\
                                                y_{n-2} \\
                                                y_{n-1} \\
                                                y_n
                                             \end{pmatrix}   \\
%
+ h^3 \begin{pmatrix*}{r}
 \frac{47}{100}     &   \frac{23}{40}   & -\frac{81}{800}   &  \frac{1}{20}     \\
 \frac{7}{27}       &   \frac{203}{729} & -\frac{25}{324}   &  \frac{65}{2187}  \\
 \frac{1399}{4200}  & -\frac{23}{168}   & \frac{783}{5600}  & -\frac{17}{2800}  \\            -\frac{109}{120}   &  \frac{61}{120}   & -\frac{81}{160}   & \frac{79}{360}
\end{pmatrix*}\begin{pmatrix}
                f_{n+1}             \\
                f_{n+2}             \\
                f_{n+\frac{8}{3}}   \\
                f_{n+3}
             \end{pmatrix} + h^3 \begin{pmatrix*}{r}
                        0 & 0 & 0 & \frac{1}{160}       \\
                        0 & 0 & 0 & \frac{31}{8748}     \\
                        0 & 0 & 0 & -\frac{13}{224}     \\
                        0 & 0 & 0 & -\frac{451}{1440}
                        \end{pmatrix*}\begin{pmatrix}
                                        f_{n-3}     \\
                                        f_{n-2}     \\
                                        f_{n-1}     \\
                                        f_n
                                     \end{pmatrix}.
    \end{multline*}
\endgroup
\end{document}

(red lines indicate text borders)

• more spaces between matrix rows is obtained by \renewcommand\arraystretch{1.3} which is put before equation (it is limited to the equation by forming group for this equation).
• instead arrays are used {pmatrix*}{r} from mathtools. it enable dhorter writing of matrices as well number in them are right aligned
• instead of aligned is used multline˛