Cause their hoping the people reading it will feel good that even if they try to help them and are unsuccessful they are still thankful to them for trying.
The messagd of writing TIA is to please try to help me any advice or tip on what the person needs help will be appreciated even if it doesn’t solve my issue so I want to thank you before to let you know that I appreciate your help even for just trying and even if it didn’t end up solving my problem.
Versus if people would not write it lots of people reading the help letter would say it sounds arrogant and the person won’t appreciate it so there’s no use in spending time helping him with tips of advice or information on his issue he needs help with.
From a psychologist
Don’t forget to Thank Hashem also whenever you get help. We are all just agents sent directly from Hashem to help others but all the help is really actually coming directly from Hashem