Rabbeinu Yonah (Shaarei Teshuvah 3:32) says it is an obligation from the Torah not to be afraid. The source is the following pasuk (Devarim 20:1): ורכב סוס וראית אויביך על למלחמה תצא כי מארץ המעלך עמך אלקיך 'ה כי מהם תירא לא ממך רב עם מצרים, "When you go out to war against your enemies, and you see horse and chariot, a people more numerous than you, you shall not be afraid of them, for Hashem, your G-d is with you Who brought you up out of the land of Mitzrayim."
The Rabbeinu Yonah writes, "This tells us that when a person sees a tzarah approaching, he should have trust in Hashem’s salvation in his heart. This is as it states (Tehillim 85:10) ישעו ליראיו קרוב אך, 'Certainly, His salvation is near to those who fear Him.' And it states (Yeshayah 51:12) ימות מאנוש ותראי את מי, 'Who are you that you fear man who will die...'"
The Rambam (Hilchos Malachim 7:15) writes, "When a person enters a war, he should trust on the Hope of Yisrael, the One who saves them at every time of tzarah. If one thinks about thoughts that make him afraid at war, he is transgressing a prohibition, for it states (Devarim 20:3) ואל תראו אל לבבכם ירך אל מפניהם תערצו ואל תחפזו, "Let your hearts not be faint; you shall not be afraid, and you shall not be alarmed, and you shall not be terrified because of them."
The Chazon Ish zt'l (Emunah u'Bitachon 2:7) writes, "Ruach hakodesh rests upon a person who has bitachon. He attains a spirit of courage that assures him that Hashem will surely help him."
Sefer Ha’Ikrim writes, “Bitachon in Hashem is greater than all praises that one can say to Hashem.” This is because one can say that Hashem can do everything, that He loves us and helps us, but it may be that his heart doesn't believe what he is saying. But, when one has bitachon, that proves he believes in all the above. He believes that Hashem loves him, cares for him, and will help those who rely on Him. Bitachon is the greatest praise because he lives and practices what he believes in.
Reb Yosef Chaim Sonnenfeld zt'l suffered many troubles in his life. (Many of his children died, among other hardships and strife.) People who knew his life story asked him how he managed to be happy. He explained to them that he attaches himself to Hashem Above, and when one is bound to Hashem, he is always happy.
He expressed it with an analogy. When there is an earthquake, people run and seek to escape. But why do they run? The earth is shaking over there as well! The solution would be to tie themselves with a strong string above because then your feet won't fall. When you are bound from above, you are safe.
A stuntman jumped to the ground from three floors high, and all the spectators were amazed. The Ropshitzer Rav zt'l took the stuntman aside and asked him how he did it. The stuntman showed him that there was a thin iron rope going down. He held on to it as he descended. It looked like he was jumping down but was using the rope for support.
The Ropshitzer Rav said that this tells us that when one is attached from above, he won't fall below.
