First go and look the types of work the gallery has, and try to imagine if your work will fit with the images or prints they are showing.

Second you have to have a fresh perspective or something unusual to show. If you are doing vegetables you better be doing them better than Weston, if you are doing portraits they should be better than Karsh or Hurrell, if you are doing flowers, hopefully they are different than and better than there lies the difficulty, you have to have a style that shows a fresh perspective and unusual or at least different qualities than what has been done so far.

Third you have to have technical expertise, you have to know what is a good print with contrast. Not everybody will like your pictures or the subjects you choose, but almost universally collectors know when a print is well done technically. I used to know a guy who wanted to be a fine arts photographer, but his prints lacked contrast, in many cases some area of the print was soft and out of focus and it was obvious it was a mistake done at the time of focusing not intentional to define the subject matter. In contrast his good friend John Charles Woods, had exquisite prints, but........

fourth, you have to be pleasant, willing not to take rejection personally, persistent and above all not have a chip on your shoulder. from the previous example neither photographer has done very well, the friend had very ordinary, derivative bad prints, and John Charles Woods, although he has written a book the two times I had the chance to meet him on a personal basis, he struck me as arrogant and that he could not be bothered with us mere mortals. He can make some of the most beautiful prints both in color or B&W I have ever seen, but you rarely hear of him, I imagine because he treats art gallery owners the same way.
IOW you have to be a good salesman, marketer and yes...have some aptitude for bullshitting.

By some twist of fate I have been lucky to meet and make friendships with photographers who at the time were unknown and now are famous and these are the qualities I noticed. They were technically excellent, they had a fresh view of things, they are persistent and thick skinned and they are very nice people to be around, willing to help and truly enjoy photography and the interaction with people that it brings.

In my experience if you have all these qualities getting into a gallery will not be hard for you, if OTOH you lack any of this, you are SOL Of course, if you can bullshit with the best of them then you might still make it, many a times I have seen the ability to market yourself be more important than the proficiency of the artist. Of course the exceptions that prove the rule are those artists who do controversial or bizarre work.