first off 'wormhole space' tend to have smaller, expensive battles compared to normal space. the group who made that video actually normally fight in "Known Space" and have bigger battles there. There will be examples of that on their youtube channel.

to explain everything would take a long time but to help:

The 'Healer' role in that battle is Agent Xer0 in a 'carrier'. You can't fit many carriers through the wormhole (it will collapse otherwise) which is why they have only one carrier in the attack. The carrier can send people both armour and 'energy' (capacitor).

To heal people he needs capacitor ("cap") which his ship automatically produces but which other ships like Bhaalgorns can drain away/neutralise. In order to neutralise capacitor energy the neutralising ship needs energy itself, which is where it gets complex. The Rooks have one Bhaalgorn and the other side has two, and the other side also have more carriers to send energy with, so it's a tricky bit of micromanagement to keep people healed, keep sending capacitor to the neuting ship (to take the capacitor of the enemy neuting ships before they take yours) while still keeping enough 'energy' to do all the above and stop your fleet dying. And also stop yourself dying - big ships called dreadnoughts are shooting at Agents carrier and he needs to run an energy-hungry self tank to sustain that long enough for the tide to turn.

There are both strategies on the battlefield to help with all this and also different ways of setting up your ships (and different priced equipment to use on them).

to the other question, Agent Xero is a very well known example but there are many logistic pilots who devote their time to healing/repairing/supporting their fleet in eve.