You can mostly make out the schematic in the photo here:
http://hipme.com/weblogs/wa5znu/20051119084647/Photo%2094.jpgAbout the only thing that would make the final smoke that fast would be a short (or at least too low
of a resistance) between the base and the 9V battery. Pull the transistor (since it let all the smoke
out already) and measure the resistance between the base connection and the positive terminal
on the battery snap - it looks like it should be around 10K. Also check from the base to the negative
terminal.
Actually, as near as I can read the schematic and parts list from that photo, it looks like there is
a 10K resistor (brown-black-orange, top of the board near the crystal) to +9V and 47K (yellow-
violet-orange by the word "Hamfest") to ground. With 10K in series with the base the current
draw of the transistor is limited to about 100mA, so the transistor could dissipate 0.9W, which
is well beyond the 625mW limit of most TO-92 plastic devices. (Note in the background of the
photo the warning not to exceed 3 seconds keydown until the transmitter is oscillating properly!)
I would expect that the current would drop once the transistor is oscillating.
What you could try is try putting the 47K resistor in place of the 10K temporarily while you make
sure it is oscillating: that limits the current draw to about 20mA. You could also insert a 12V
light bulb or 470 ohm resistor in series with the transistor collector (or in series with the key)
to reduce the power dissipation while you are experimenting. I don't see any obvious adjustment
or tuning to get it to oscillate, but that probably is in the instructions somewhere.