Can someone please teach me how to diagnose what's wrong with this circuit board it's for the led lights for my tail light

 

I'd start by making sure it's getting power and/or a signal. In other words there may be nothing wrong with the board itself. Do you have a known-good board you can sub in for it? Or, can you place this one on a known-working circuit? Do you have a multimeter for probing for voltages? You want to narrow in on the problem, but you do that by collecting data.

 

if I connect 3 red leds and a resistor will that work and how long u think it will last to replace the one in the picture. For the tail light for my car. The thing is the dealer don't sell that piece I have to buy the entire tail light and it cost $290

 

As long as the rest of the car can't tell the difference it'll probably last longer than you will or the car for that matter. Problem is that cars nowadays have all sorts of auxiliary pieces that tell it things about the health of the circuitry. It will record what it considers shenanigans and rat you out to the dealer who may void your warranty for unauthorized repairs. Just be glad it's not a John Deere tractor. Owners of those machines have to rely on Ukrainian hackers to fix their tractors, because the dealership charges thousands for replacement boards.

 

To check the LEDs the simple method requires a regulated power supply that can be set to low voltages below 5 volts, and supply at least 50 milliamps. Use the diode-test setting of a digital multimeter to determine which side of each LED gets the positive voltage, and then power each LED to verify that it lights up. You do not need to provide the maximum current to verify that the LED is OK. I also see what looks like a surface mount resistor, check with the ohm meter that it is not open. And I think that there is a chip capacitor as well.
But my guess is that it is that small IC that has failed and replacements for those are usually not available.