What is the meaning of |H(jw)| concerning the transfer function
A transfer function takes a complex number input (or in this case, pure imaginary \$j \omega\$), and produces a complex number output.

The "magnitude" of \$H\$ is simply the complex absolute value:

\begin{gather}
|H(j \omega)| = \sqrt{\mathrm{real}(H(j \omega))^2 + \mathrm{imag}(H(j \omega))^2}
\end{gather}

As far as a physical meaning, consider an input signal with an amplitude \$V_0\$ and frequency \$\omega\$. \$H(j \omega)\$ "transforms" this input signal into an output signal, with final amplitude \$V_1 = |H(j \omega)| V_0\$, but it also can shift the phase of the output signal with respect to the input signal.

To summarize, the two plots of the Bode plot are:


\$|H(j \omega)|\$, which tells you the ratio of input and output amplitudes (the "gain", or "attenuation")
\$\tan^{-1}\left(\frac{\mathrm{imag}(H(j \omega))}{\mathrm{real}(H(j \omega))}\right)\$, which tells you the phase shift (note: this formulation is only accurate in the 1st quadrant; see atan2 for the definition which extends to the entire complex plane).