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After Connor MacLeod beheads the Kurgan, he goes through some kind of event, as had other Immortals after winning battles, but this time it is accompanied by various animations of what appear to be "evil spirits" attacking him.

I have always interpreted this to be that the victor is absorbing the essence of the immortal they killed, and in this case, Connor is absorbing the evil essence of the Kurgan, but there is nothing in the film to suggest this, though I vaguely remember it working this way in the TV show.

UPDATE

OK, found something that indicates where the dragon image MAY have come from. Wikipedia (without citation) states the following in "Alternate releases":

Proposed duel in the ending

In the scene following Connor beheading the Kurgan, Mulcahy had originally envisioned an animated dragon with the Kurgan's battle helmet emerging from his decapitated body and challenging Connor again. Only after Connor had defeated this ghost-dragon would he have received the final Quickening and subsequent Prize. This idea was eventually cut due to budget constraints.

If you check the images, they DID have the energy look at times like some sort of monster/animal spirit. Mostly looking like Chinese Dragons (the screenshot I found isn't good for that but I re-watched the scene and it shows the long dragon body before that).

ORIGINAL ANSWER

That's Quickening.

There's really no good canon explanation of details of it in the first movie itself. There's no "evil spirits" or "evil essence". Just energy/spirit/whatever.

Note that most of the Quickening it was just pure energy:

As the canon stands (alternate release info above nonwithstanding) there's no information leading one to believe that the dragon image had any deep meaning aside from "this is the special Quickening that gave MacLeod the Prize".

Now, I looked at the script:

1. The first cut of the script (didn't even have the name Nash yet - it was Taupin; and Kurgan was The Knight) didn't have anything useful (search for the section starting with "And CUTS the Knight's head off"). No mention of spirits/shapes.



2. 
   

   Next we have 2/6/1985 script. This is final, but NOT the director's cut - for example the start has MSG hockey game instead of wrestling. The needed piece (search for "He cuts off the Kurgan's head") has some info but the only "shapes" are MacLeod taking shapes of those Immortals he killed:

   
   
   

   
   

   Transfixed, Brenda watches a sparkling energy leave the corpse and engulf MacLeod.
   
   MacLeod glows all hues of the rainbow.
   
   His hair stands on end. Color erupts from his eyes.
   
   He's like a roman candle against the waves. The WIND HOWLS.

   
   
   

   Brenda watches MacLeod's body transform into many different people. His voice ECHOES over the surf.
   MACLEOD:
   Everything's alive. The Quickening overpowers me.
   
   (becoming Ramirez) All resistance is gone. I am generations being born and dying.
   
   (becoming Fasil) I am night air, breathing.
   
   (becoming unknown people) The life-force smothers me.
   
   (becoming the Kurgan) I am all of them.
   
   (becoming himself) I can feel everything

   
   
   

   She's terrified. His form shifts and changes before her.

   
   

   
   


3. 
   

   Finally, we have Garry Kilworth novelization, again not mentioning any animations:

   
   
   

   
   

   Windows began to implode. The whole studio shook and Brenda screamed,cowering in a corner, as a blizzard of glass filled the room.

   
   
   

   The energy was now visible in the air, flowing through the storm of powdered glass. There was a swirling, circular motion to the energy, as it formed itself into a maelstrom, a vortex. MacLeod felt the power blazing through him, burning through his veins, searing through his spirit. His whole being was alive, glowing with the reward of his success.
   He felt light. He was lifted from his feet through the eye of the vortex: lifted high above the floor, hidden from Brenda’s view by the swirling white dust of the glass. The pain and the pleasure mingled. His brain was full of colours, full of light. He could hear himself screaming, the wind and the rain of energy still lashing into his body. He felt god-like, yet undergoing the punishment of a god. Then he blacked out.

   
   

   
   

The only "good/evil" thing that is known was NOT in the movie, but in the series (and didn't involve spirits either). To quote Wiki:

On rare occasions, the Quickening overwhelms the personality of the Immortal, even turning a good Immortal evil. This occurrence is known as a Dark Quickening in the series version (reference: Episode "The Sea Witch," Bonus Material, William Panzer's interview, in Highlander: The Series (season 1) (DVD, Davis-Panzer Productions, Inc., 2001), disk 3.)

(there was apparently a reverse process as well, e.g. Sean Burns Quickening got Dunkin to become "less evil" after a Dark Quickening).