I can hear through my headset but I can't talk

Check to see that your mic is plugged in to your headset correctly as there should be a white circle that appears around your gamertag when you talk.

The issue you are having is your controllers software is out of date when you plug in your controller and start a party and there should be an option in the party menu to update your controllers software

busca actualizar los controladores si aún no lo ha hecho.

The first option is to check for controller updates if you haven't already done so. Hopefully there's an update pending, and that fixes the headset issues you're having. You can do so via the Xbox Accessories app on your console.

Greetings.

As mentioned below, check for controller updates. If this option does not work, then check to make sure the mute button is off or the your settings in the Xbox itself. Believe it or not, self muting happens alot

If all else fails, try borrowing another controller and see if you have the same issue. If still the same, I would check the headset compatibility with the xbox controller and/or contact support