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For example, say I have a Mandibuzz with Dark Pulse and Nasty Plot. I use Nasty Plot to raise Sp. Atk. by two stages before using Dark Pulse. What exactly does the stage raising do/mean?

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It raises your special attack. For example, its SpA is 100. Since it raises by two stages, the SpA becomes 200 altho if you check the summary, it won't change. So basically one stage = half of the original stat. Since the example is 100, one stage is fifty, two is one hundred, three is one hundred fifty, etc. I'm not sure if this is how the stages work if the stats are reduced. What I do know is that it is either a quarter of the original stat or a third. I'm not sure about this so I'll just leave it as a comment

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Physical attack, physical defense, special attack, special defense, and speed get multiplied by a certain amount depending on the stage. Accuracy, evasion, and critical hit chance also have stages, but they work differently.
-6: *2/8
-5: *2/7
-4: *2/6
-3: *2/5
-2: *2/4
-1: *2/3
0: *2/2
+1: *3/2
+2: *4/2
+3: *5/2
+4: *6/2
+5: *7/2
+6: *8/2
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For standard stats, each single boost counts as a 1.5× boost to the raw stat. Each boost is additive, not multiplicative. This means being at +2 means a 2× boost instead of a 2.25×. Maxed out at +6 boosts means a 4× boost to the raw stat.

For lowering, the stat is 1/the boost, so -1 is 1/1.5× or 2/3×. -6 is 1/4×.

Accuracy and Evasion follow the same principle, but add 1.33× instead.

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Being at +2 Sp. Attack means your special attack is doubled.