Three aspects of Edgars disguise as poor Tom are first, that he is mad or insane. He pretends he escaped from Bedlam, an insane asylum. Second, he pretends to be a beggar, with nothing more than an old blanket to cover himself with against the elements. Finally, he tells Lear and the others that he was once a proud servant who had an affair with his mistress. He claims to have been dishonest, stealthy, greedy, and lazy.
All of these personas distance him greatly from the real Edgar his father or Lear would recognize. As eNotes point out, the insane were invisible in this society, so Lear and Gloucester were not likely to examine his appearance too closely. The same could be said of beggars. Finally, the former servant that Edgar describes is so different from Edgar himself in character that it is unlikely anyone would associate him with that figure.
Edgar has done a good job of thinking through his disguise.