Condition for absolute stability
The other posters make references to the misconception that if the gain is larger than 1 when the phase shift has reached 180 degrees in the open loop, the system will oscillate when the loop is closed. Let me provide a counter example. Consider the open loop transfer function:
$$L = \frac{10 (s + 1)^2}{s^3}$$
Here is the bode plot:

Clearly the system should oscillate when the loop is closed, right? Well, not quite:
$$H = \frac{L}{L + 1} = \frac{10(s+1)^2}{s^3 + 10(s+1)^2}$$
The poles are:

Looks stable to me.

The thing is there is not an easy explanation. Here is something from mit. To quote the first sentence: "The  Barkhausen Stability Criterion is simple, intuitive, and wrong."

The truth is you would want to look at Nyquist, and the theory behind it, i.e. Complex analysis, winding numbers etc., in order to understand what is going on.