How does an IR LED turn on and how can I test to see if its not broken?
An IR LED is a "regular LED". It's a diode that emits light.

You hence use it like any other LED: You get a datasheet that says how much forward current they are typically used with (e.g. 10 mA), what the forward voltage is (e.g. 1.6 V), and then you put them in series with a series resistor to your voltage source. I'm not going to go into detail about series resistors here, because there's literally thousands of explanations and calculators for LED series resistors out there.

You might be confusing an IR LED with an actual IR transmitter integrated circuit, which contains an IR LED, but also other things like a digital packetizer, and a power supply.

But: if that's the case, we have no idea what IC you're talking about, so we couldn't answer that.

EDIT after you added a product page link:

This looks like LEDs. So, this is just a plain IR LED.

By the way, amazon's not the place to buy electronic components. You wouldn't buy car tires in a vegetable shop, either. Go for one of the large electronic distributors, like Arrow or Digikey. You'll notice that when you buy from professionals, components will come with datasheets that actually define what they do.