TIAs typically happen because a blood clot gets lodged in an artery that supplies blood to the brain. Without regular blood flow, your brain is starved for oxygen and can't work like it normally does.
That's why you get symptoms like muscle weakness or slurred speech. It'd be like having a clogged fuel line in your car. Your engine can't run if it's not getting gas.
You can also get a TIA if so much plaque builds up in an artery that it severely limits blood flow to the brain, like a clot.