Why I find a real pole of a transfer function leads to an imaginary frequency and a imaginery pole of a transfer function leads to a real frequency?
You are mixing Laplace-domain transfer functions with Fourier-domain transfer functions.
Either you start with the Laplace version:
$$H(s) = \frac{\omega_1}{s + \omega_1}$$
and the system response to a sinusoid \$x(t) = \cos \omega t\$ is found by evaluating \$H(s)\$ at \$s = j\omega\$.
Or you use the somewhat more obscure Fourier version:
$$H(\omega) = \frac{\omega_1}{j\omega + \omega_1}$$
and the system response to a sinusoid \$x(t) = \cos \omega t\$ is found by evaluating \$H(\omega)\$ at, well, \$\omega\$.
There's some mathematical 'i's and 't's that get dotted and crossed differently with Laplace vs. Fourier, so there's a few cases where one is valid and the other isn't.  Mostly, if you have to ask, they're equivalent ways of saying the same thing.