Having spent some time with my Tanach over the weekend I suddenly realized that Tehillim is a part of Tanach along with every other sefer. I was wondering when did Tehillim become the book that people recite when someone is sick or if they need some kind of yeshua.
The Leviim in the Miqdash would recite portions of Tehillim and as a remembrance of this, we continue to recite the shir shel yom drawn from passages of Tehillim. This indicates that drawing upon Tehillim as an aspect of avodah has roots at least as early as the 2nd Temple. As a function of exile, the avodah is replaced with tefillah. Other than the berakhoth (in the amidah or otherwise) and the biblical obligation of shema, most of our prayers consist of Tehillim. Most of Pesuqe d'Zimra, Ashrei (read thrice daily), Tahanun, Lamnaseah, shir shel yom, most of Qabbalath Shabboth, Hallel, etc. are all comprised of Tehillim. The composition of Tehillim being mainly songs/hymns of praise and affirmation of divine salvation, readily lends itself to such use, whether communally or individually.
Tehillim was written by Dovid Hamelech, with Ru'ach Hakodesh. He depicts many sufferings that he went through and adds how much He feels protected by Hashem, and how Hashem helps him achieve higher levels (of faith and of observance). He also talks of national challenges and how Hashem gives us dark times and healings. As such people find solace and faith as they recite or listen to the words.