Wrapping text in a table cell

I made three changes to your table:

1) I added \centering

2) I changed the width of the 4 right-side columns to 0.2\textwidth..., and

3) I placed the headers each in its own \Longstack. This required three additional tweaks in the preamble: add the stackengine package; tell the package to use \# as the end-of-line, since tabular is already using \\ (note that the latest stackengine version 4.00 can use \\ end-of-line delimiters, even nested inside a tabular); and tell the package to set 12pt between baselines of the stack lines, since tabular zeroes out the value of \baselineskip, which is otherwise the default long-stack gap. Note that you can give the \Longstack a [l] or [r] optional argument to change it from the default [c] alignment.

\documentclass{article}
\usepackage{booktabs, threeparttable, stackengine}
\usepackage{array}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}}
\setstackEOL{\#}
\setstackgap{L}{12pt}
\begin{document}
\begin{table}[ht]
\centering
\begin{threeparttable}
\caption{Minimum Requirements for Automatic Readmission into the Science Faculty}
\label{table:sci}
\begin{tabular}{@{}p{0.18\textwidth}*{4}{L{\dimexpr0.20\textwidth-2\tabcolsep\relax}}@{}}
\toprule
& \multicolumn{2}{c}{\bfseries \Longstack{Cumulative Number\# of Courses Required\# to be Passed}} &
\multicolumn{2}{c}{\bfseries \Longstack{Cumulative Number of\# Senior Courses Required\# to be Passed}}  \\
\cmidrule(l){2-3} \cmidrule(l){4-5}
& BSc & EDP & BSc & EDP  \\
\midrule
First-year & 2 & 2 & --- & ---  \\
Second-year & 7 & 6 & --- & --- \\
Third-year & 11 & 10 & 3 & 2 \\
Fourth-year & 15 & 14 & 6 & 5 \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}
\end{document}

Don't use c in \multicolumn{2}{c}{... but

 \multicolumn{2}{L{\dimexpr0.3\linewidth-8\tabcolsep\relax}}{...

Full code:

\documentclass{article}
\usepackage{booktabs, threeparttable}
\usepackage{array}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}}


\begin{document}
\begin{table}[H]
\begin{threeparttable}
\caption{Minimum Requirements for Automatic Readmission into the Science Faculty}
\label{table:sci}
\begin{tabular}{@{}p{0.18\textwidth}*{4}{L{\dimexpr0.3\linewidth-8\tabcolsep\relax}}@{}}
\toprule
& \multicolumn{2}{L{\dimexpr0.4\linewidth-4\tabcolsep\relax}}{\bfseries Cumulative Number of Courses Required to be Passed} &
\multicolumn{2}{L{\dimexpr0.4\linewidth-4\tabcolsep\relax}}{\bfseries Cumulative Number of Senior Courses Required to be Passed}  \\
\cmidrule(l){2-3} \cmidrule(l){4-5}
& BSc & EDP & BSc & EDP  \\
\midrule
First-year & 2 & 2 & --- & ---  \\
Second-year & 7 & 6 & --- & --- \\
Third-year & 11 & 10 & 3 & 2 \\
Fourth-year & 15 & 14 & 6 & 5 \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}
\end{document}