Why should one say Tehillim for the sick?

Aneinu

One fulfills the Torah Mitzvah of (וְאָהַבְתָּ לְרֵעֲךָ כָּמוֹךָ) v’ahavta larayacha kamocha, which emphasizes the importance of loving one’s fellow human being. Source: Chashukei Chemed Brachos 12b. The act of reciting Tehillim is seen as an attempt to reverse any (גזר דין) strict decree placed upon the sick person and instead invoke (רחמי שמים) mercy from heaven.

וְאָהַבְתָּ לְרֵעֲךָ כָּמוֹךָ אֲנִי ה: Kamocha (כָּמוֹךָ) is gematriah 86 which is equivalent to the Name of Hashem that represents strict justice ( אלהים ) Elokim. If one focuses on loving one’s fellow – specifically, this individual, then ‘ani Hashem’ (using the Name that represents Divine mercy יהוה )

Also, The Gemara (Brachos 12a) tells us that “Anyone who is able to pray for his friend and does not is called a חוֹטֵא .”

If you would like to know which kapittel to say: Tehillim for the sick.