The Bode Plot is typically used to display the open loop magnitude and phase response, for which we can assess stability in many cases (not all). The stability criteria that the phase is less than -180 degrees when the gain crossing is at 0 dB.
The relationship between closed transfer function (or "closed loop gain") and the open loop transfer function ("open loop gain") is given as:
$$G_{CL}(s) = \frac{G_F(s)}{1+G_{OL}(s)}$$
Where $G_{CL}(s)$ is the closed loop gain, $G_F(s)$ is the forward gain from the input to output of the closed loop system, and $G_{OL}(s)$ is the open loop gain which is the cascade of all the system components inside the loop, assuming a negative feedback which is not included as part of the open loop gain (hence the instability by having positive gain with feedback of 180 degrees which combined with the negative feedback causes the in phase positive gain that leads to instability).
The denominator of the closed loop transfer function $1+G_{OL}(s)$ is called the characteristic equation and the roots of this (the closed loop poles) ultimately tell us if the system is stable or not. A big utility of the open loop gain alone and the use of the Bode Plot (and Nyquist plot) to assess stability is that it is something we can directly measure (in the case of stable open loop systems) and we can derive the Bode Plot even when an actual transfer function cannot be established such as the case of time delays in a continuous time system and other cases involving transcendental equations that can't be described with polynomials.
This may make your head spin but gives interesting insight: for any ratio of polynomials $H(s)$, the "poles" are the values for $s$ that make $H(s)$ go to infinity. The "zeros" are the values for $s$ that make $H(s)$ go to zero. For example the transfer function $H(s)=(s+1)/(s+2)$ has a zero at $s= -1$ and a pole at $s= -2$. Note how the open loop gain $G_{OL}(s)$ is itself a transfer function with its own poles and zeros. However it ends up in the denominator of the closed loop system. However, any value of $s$ that makes $G_{OL}(s)$ go to infinity will still make the characteristic equation $1+ G_{OL}(s)$ go to infinity! So the poles (which would do that) in the open loop transfer function, are the zeros in the closed loop transfer function, and those never move their location. However the zeros in the open loop transfer function are not the roots of the characteristic equation (they are the values of s that make the open loop transfer function go to zero, but $1+G_{OL}(s)$ would go to $1$ so it is not a root! Therefore the zeros in the open loop gain are not the poles. The whole root locus thing (we see where the closed loop poles are as we vary a gain constant $K$) is done by modifying the characteristic equation to be $1+K G_{OL}(s)$, and here we get insight into how the poles in the closed loop system move to toward the zeros as we increase the loop gain $K$. For $K$ very large, $G_{OL}(s)$ dominates the denominator, and therefore the closed loop poles will get increasingly close to the open loop zeros (which as we mentioned are also the closed loop zeros)! Note that for systems that have "more poles than zeros", these actual refer to finite poles and zeros. All systems have the same number of poles and zeros, and the zeros we don't see are out at infinity. So in those cases you will see the poles in the root locus going out toward infinity to find their zero.
