LaTeX blank after number and before text

Use p{width} for your column specifiers instead of l/r/c.

\begin{tabular}{|p{1cm}|p{3cm}|}
  This text will be wrapped & Some more text \\
\end{tabular}

EDIT: (based on the comments)

\begin{table}[ht]
    \centering
    \begin{tabular}{p{0.35\linewidth} | p{0.6\linewidth}}
      Column 1  & Column2 \\ \hline
      This text will be wrapped & Some more text \\
      Some text here & This text maybe wrapped here if its tooooo long \\
    \end{tabular}
    \caption{Caption}
    \label{tab:my_label}
\end{table}

we get:

\mbox is the simplest answer. Regarding the update:

TeX prefers overlong lines to adding too much space between words on a line; I think the idea is that you will notice the lines that extend into the margin (and the black boxes it inserts after such lines), and will have a chance to revise the contents, whereas if there was too much space, you might not notice it.

Use \sloppy or \begin{sloppypar}...\end{sloppypar} to adjust this behavior, at least a little. Another possibility is \raggedright (or \begin{raggedright}...\end{raggedright}).