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The UI for Incarnon Melee could honestly use a bit of work.  All the pieces are there, but they're harder to parse than their primary and secondary equivalents.  Setting that aside, I can explain how it works to you.
Different Incarnon weapons have different combo requirements to transform into their Incarnon state.  Some require 5x combo, some require 6x combo, and there are some that have optional upgrades that drop it down to 3x combo.  You can tell how much combo any given Incarnon weapon requires by going to your Arsenal, equipping the weapon, and then on the right side of the screen you will see the weapon's upgrade nodes.  Hover over the top-most node and it will tell you how much combo you need.
In-mission, you can see when you have enough combo to transform because your combo meter will start glowing; this will be very hard to miss if you're looking for it as it's a vivid effect.  Once you see this effect, you can press the heavy attack button, and doing a heavy attack will transform your weapon for a certain amount of time, after which it will revert back to its normal state.  When your Incarnon weapon is in its transformed state, one of your buffs will be the amount of time you have left in Incarnon state.  If you can't tell which one it is, you can pause the game and cursor over your buffs to get pop-up information about them.
Hope this helps!

The way damage is calculated in Warframe is incredibly unintuitive!  Rather than bang your head against a wall trying to reverse engineer it, I'd recommend the video I'm linking at the end of this post, which in my opinion explains how all the numbers fit into Warframe's magical equations quite well.
That said, I can explain what you're seeing in this instance.  In your test, you're using 3 total mods that get sorted into 2 different categories
Base Damage mods Elemental damage mods In terms of calculating total damage, all mods of a specific type get added up.  So we get the following:
BaseDamageModMultiplier = 1 + Hornet Strike + Pistol Amp = 1 + 2.20 + 0.27 = 3.47 ElementalDamageModMultiplier = 1 + Primed Heated Charge = 1 + 1.5 = 2.5   (assuming the mod isn't maxed) Now we calculate the total damage by multiplying these values against your weapon's base damage:
WeaponBaseDamage * BaseDamageModMultiplier * ElementalDamageModMultiplier = TotalDamage So... 325 * 3.47 * 2.5 = TotalDamage = 2819.375 I hope this helps!
 
Here's the video, which explains other things I didn't cover here like how Multishot works into the equation:
 

Yes, that's correct! :)