[kekie]

The very first step - where the Na donates a electron and one of the C-C pi bonds breaks - I don't understand why it happens.
I've googled plenty, to no avail. Every resource I've found explaining the mechanism shows it happening, but doesn't explain why.

[AlphaScent]

the sodium donates an electron to give the.carbanion.  it  does this because of the nature (reactivity) pf sodium,  especially in liquid ammonia.  have you.ever done this?  I do it with lithiumand there is a bkue color which shows the solvation of an electron.  I am on my phone so my answers are limited. 

look at.where sodium is.on.the periodic table, what.are its electonic properties. this will give a better idea of.the why to this reaction

[kekie]

I get that sodium really wants to give its electron up (like in table salt) but why does the bond break the way it does? And I can't imagine that the carbon is much of a electrophile...

[orgopete]

There two factors, first carbon is in the middle and you can find examples of it being an acid (acetylene or cyanide) or as an electron donor (and becoming positively charged), so it could do either. Secondly, an sp2 and more so, an sp-carbon is electron withdrawing. You can convince yourself by the increased acidity going from ethane, ethylene, and acetylene. So you might expect an acetylene could accept electrons from a sufficiently strong donor.