Baha'is believe because they heard the call of Baha'u'llah and were compelled to investigate and discover the truth for themselves. Now, some people are exceedingly pure-hearted and immediately recognize the Manifestation of God just by hearing a few things about Him and His message to humanity. However, in this age God does not require blind faith. God is asking humanity to be mature and independently investigate the claims of Baha'u'llah.
Interestingly, most people who humbly investigate can't help but believe because His writings prove that He is the promised one. In the time of Jesus very few people believed and that's why when given the choice between freeing Barabbas, a known murderer and thief, or freeing Jesus whom some called the Son of God, the crowd chose to crucify Jesus, and instead of seeking to know the truth, they freed the murderer and thief.
In summary, the reason Baha'is believe is because when they learn about Baha'u'llah, it can metaphorically be said that they "hear the voice" of the shepherd and answer the call of God.
Baha'u'llah revealed a mystical book called the Seven Valleys. It tells the seven stages that a seeker may traverse to find the object of His quest. And He says that sometimes the seeker will swift at the blink of an eye reach his objective.
Here is what Abdu'l-Baha says about the first principle of Baha'u'llah which is search for truth.
In short, the seeker has to be persistent because, in most cases, being an earnest true seeker isn't easy to do because we all carry a lot of baggage from our culture, our training, our education, our religion, our friends and family, and most of this baggage is pretty heavy and won't help us find truth.
The truth is Baha'u'llah lived for forty years in exile and as a prisoner, and what He experienced convinced Him of the plight of humanity; a burden we all share, and that means we have to be diligent and earnest in our quest for truth:
"... the people are wandering in the paths of delusion, bereft of discernment to see God with their own eyes, or hear His Melody with their own ears. Thus have We found them, as thou also dost witness.
"Thus have their superstitions become veils between them and their own hearts and kept them from the path of God, the Exalted, the Great."
"O people, if ye deny these verses, by what proof have ye believed in God?"