How can I amplify 868MHz transmission done with a wire antenna?
Instead of messing around with a RF power amplifier, which may not be legal, I'd spend the effort to make the existing system work better.

Probably the transmitter and receiver are too far apart, or have obstacles between them that interfere with the radio waves.  Since the garage door is where it is, maybe you can move the receiver in the house.  If not, maybe you can make it pick up the garage door transmitter better by running a wire from it to nearer the transmitter.

The wavelength is 350 mm or 14 inches, so a directional receiving antenna isn't out of the question.  You can do whatever you want on the receiving side without violating any RF emissions laws.  As soon as you touch the electronics in the transmitter, the unit is no longer certified, and you're on your own.  It is quite unlikely that anyone will come after a private person with a single garage door opener, but if you happen to interfere with something important you could get into trouble.

Both installations are fixed, so you should be able to run some wires.  Even if you can't add wires to where the receiver is, you could still make a passive antenna extender.  Put a 1/2 wavelength dipole on each end of UHF twinlead.  Get one end a few feet from the transmitter, and the other a few feet from the receiver.  Look at how the antenna on the circuit board is oriented, and adjust the orientation of the dipole accordingly.