Recently went through the process this past cycle. According to quite a few program directors, it is definitely getting more competitive each year. This year, there were several people that did not get positions. I had program directors tell me that they got upwards of 40 applications for only 2 spots. Granted, a lot of applicants probably "shot-gunned" the process and applied to a high number of programs (I certainly did this because I am a DO). I would recommend for any FMGs or DO to apply to a higher number of programs. In addition, CT Anesthesia will be implementing the SF Match for this next cycle. I would imagine this would mean that programs will have to interview more candidates, but am unsure.
As for what the programs look for...it is hard to say as every PD might be interested in something else. Some will favor board scores, ITE/AKT scores. Some will put heavy weight on LORs. Some will go searching for what research experience you have.
One thing that was almost universally recommended was that you should only look for accredited fellowship programs, otherwise you could not get TEE certification.
Overall, applied to 15 programs and got 5 interviews and subsequently got 2 offers. Luckily, one of those offers was to my #1 choice, so I am ecstatic.
Hope this helps. Best of luck during the match.