How to continue the framed text box on multiple pages?

Use the mdframed package which extends framed (and is rather configurable compared to its ancestor):

\documentclass{article}

\usepackage{mdframed}
\usepackage{lipsum}

\begin{document}

\begin{mdframed}
  \lipsum
\end{mdframed}

\end{document}

An alternative package to mdframed is tcolorbox. Among some other interesting features, includes breakable framed boxes.

Using default style a simple example could be:

\documentclass{article}
\usepackage[most]{tcolorbox}
\usepackage{lipsum}

\begin{document}

\vspace*{15cm}
\begin{tcolorbox}[breakable, enhanced]
\lipsum[1]
\end{tcolorbox}

\end{document}

and the result: