If z is negative in period t, how would you think about that in nonmathematic terms? Do machines suddenly decide to temporally become less productive?
Can someone please explain to me wtf a negative shock on technology is?

I agree in terms of growth rates, but RBC includes a technology shock in the production function.
Lower growth rate of TFP relative to trend. It's not so hard to understand except if you are an applied labor economist for whom the world outside LATE seems so mysterious.

The income identity
Y = W+P
can mathematically be transformed into
dlog(y) = [a*dlog(w)+(1a)*dlog(p)]+a*dlog(L)+(1a)*dlog(K)w and p are relativ factor income/costs, a is labor share.
The term in the square brackets is your „TFP“, as measured with nipa/sna data.
Now you can proceed with reasoning about the behaviour and driving forces of the wage and profit rate during the cycle.

And on top of that the Fed knew this was the real reason for the recession and had to use monetary policy to combat this negative technology shock.
It wasn't the housing and offbalance sheet derivatives markets that caused the financial crisis it was the iPhone causing a huge shock.It's when Steve Jobs introduced the iPhone and made dumb people into dumb zombies staring at their iphones all day. It's a shock that persisted since 2007.

A pervasive problem across economics is misleading terminology and aggressive interpretation of residual terms that pickup any specification error, measurement error, etc....
The simple model Y=A*f(K,L) has (imperfectly measured) observables of output Y, capital K, and labor L. Assume a production f, and then A is a residual term to make everything up.
An aggressive interpretation is that changes in A are technology shocks, but specification error and measurement error will also show up in A! Blowing up a bunch of capital without recording it in K_t will get picked up as a negative shock to A_t.