Although I will be adding my own thoughts here, please bear in mind that this is someone that prefers to pilot carriers/supercarriers as the flagship in a carrier centric fleet. Honestly, I was pretty excited when 0.8 rolled around and Alex announced the new mechanics for fighter control that wasn't "Mass tons of fighters with a few hangers and watch them wolfpack them down."
Generally, fighters do better against frigates and destroyers while bombers are better against cruisers, capitals, and starbases. Another unmentioned reason that fighters are strong are because of the AIs tendency to deploy all small ships first before the larger ones. This leaves nothing but separate frigate and destroyer fodder to be quickly run through before the less DP efficient enemy caps and cruisers show up lacking any support and getting worn down.
The video also shows the main advantage of fighters, the concentration of force with little risk. Early on you took down the level 2 station by concentrating the fighters immediately on it prior to the enemy frigates showing up, and as stated before frigates are fodder to fighters and quickly disposed of by the high-end performing sparks. This meant that for the first few minutes you were able to safely concentrate 160 DP worth of firepower on the station. Some fighters would be destroyed of course, but it means little to losing frigates or destroyers.
In this case too, as noted earlier, pirates tend to be very poor in dealing with fighters, let alone the extremely high end and dangerous shielded 5x Burst Laser sparks. Pirates lack good anti-fighter weaponry, either have no or poor shields with little coverage, lack good options for bigger ships aside from Falcon(P), and their officers tend to be few and low in number, meaning less chance for 'Advanced Countermeasures 3'.
A high tech space fortress might be the better contender against the spark drover due to having better, dedicated shield modules and heavy mine spam. It won't kill the drovers not being mobile, but if it isn't slowly getting worn down it should force a stalemate and retreat. High tech would also be more likely to make use of dooms and omens which are much more effective at anti-fighter.
Although mods were involved, in 0.8 I used an II Matriarch loaded with Wanzers and was able to blow through most opponent fleets with the exception of the Templars, who were strong due to their priwen bursts and the high frequency of 'Advanced Countermeasures' from all level 20 officers and DME who have a lot of EMP, which is better against low numbers of strong fighters admittedly.
Generally, PD weapons are not so good against fighters though you'd think so by designation. Since all weapons can target fighters it works a bit differently, HIL for instance can be manually swept over an initial fighter wave with its strong HE. Swarmers can be massed rather than harpoons for good anti-fighter coverage for a time and the first wave at least. And don't dismiss Advanced Countermeasures 3. AC 3 with good autofire accuracy can single-handedly wipe fighter squads with token defenses or weapons, 50% damage is no joke. Usually when piloting a carrier, I can immediately tell when an enemy ship has AC 3 regardless of class by the amount of casualties they inflict. It won't save a frigate, but for a cruiser and capital it makes a huge difference in whether I'll be able to beat them down or if they completely no-sell my carrier and its a very good skill to give to officers if one is inclined to go warship heavy as it counters fighters surprisingly well. Using the templar example above, AC 3 bulwarks were almost entirely untouchable with priwen bursts, and the paladin and archbishop would kill them before they even got close.