The tube bias adjusts the operating point of the tube. What you would really like is a tube that is linear, which means the output signal increases exactly the same as the input signal. If you input signal goes up by lets say 1 volt, and that causes your output voltage to go up by 10 volts. (Just example numbers) Then if you increase the input signal to 2 volts the output should be 20 volts. That would be a linear gain. Unfortunately for hi-fi types and fortunately for guitar types, it doesn't work that way. When you increase the input to 2V the output may only go to 17 volts. That's a non-linear response, otherwise known as distortion, or clipping, or whatever.
When you adjust the bias you will move the voltage that comes out the output, how much gain there is, and also where and how much the non-linear distortion starts to happen.
Lets take the example above. If we increase the bias on that tube amp, the the 1V might give 15 volts output, but a 2V input might only give you 17. It starts distorting much earlier in terms of the output.
The same thing happens at the other end with low signals. If the bias is too low, then you get distortion at the other end, at say 1/10V input voltage.
They try to balance the tube so it gives as clean a signal as possible over the full range of the amp.
By running the bias really high you can make the amp have tons of distortion even at moderate output levels and even more when you crank it up. Unfortunately, the tubes will get very hot and it will shorten their life. I read somewhere that EVH had his tubes all biased so hot that they were only good for about 1 show. I guess if you are EVH you can afford that.
The sag is a flaw in the power supply rectifier tubes from back in the day. They were not really very good. A lot of the power was lost in the rectifier tube itself. When you played a really loud chord or something, then the amp needs more power, but the rectifier can't supply it, so the amp is starved for power. This is sometimes called compression. It compresses the dynamics of the guitar signal, kind of like a pedal will do. It lowers the amplitude of the peak signal, simply because the amp can't get the power from the rectifier. You can eliminate this by increasing the size of the power transformer and changing the rectifier tube to a solid state rectifier (diodes). Probably a larger filter cap as well. But that is one of the effects that people like about tube amps.
That's the engineering perspective, at least as I understand it.
I don't really like a lot of sag, or distortion. But I do like tube amps, so go figure.