Excellent answer from Shaykh Zoheir Esmail, I also wanted to add that there is some indication in narrations that having faith and wilayah in the Imams (A) does develop this capacity to some degree. For instance, it is related, from Imam al-Sadiq (A), ‘Our followers (shi'a) have have four eyes: two eyes in the head and two eyes in the heart. Actually, all people have these, but Allah opened your sight.'
If one goes through what is found in the books of narrations, there are some pointers here and there on this topic. For instance, you may be familiar with the saying attributed to Imam 'Ali (A) saying 'lower your eyes and you will see wonders'. While this is often discussed in the context of having a chaste gaze in society, it can be taken on a deeper level to mean avoiding distracting one's self with the glitter of the life of this world to be able to understand beyond it.
It is my view (and by that, I mean it is my personal view and may or may not be the view of others) that both the formally prescribed practices of Twelver Shi'ism (such as prescribed du'as or a'mal as well as ziyarat to the holy sites) as well as the traditional practices (especially relating to majalis and azadari) do both facilitate a stronger sense of spiritual vision and understanding if one takes them seriously. After all, it is really not uncommon for Shi'is to experience, or at least say they experience, dreams, visions, miracles, etc, of Ahl al-Bayt when participating in these events.
One can always pray for what one seeks, whatever it happens to be in life, including spiritual vision. One could recite one's own prayers in one's own words, or there are some snippets of du'as from Ahl al-Bayt (A) that could be considered appropriate such as from Munajat Sha'baniyyah.
Beyond that, many things in life are not really learned by books. One can get inspiration or some tips from books, but many arts really are passed on through a person to person basis. For instance, few people solely learn cooking or swimming from books, even though one can get tips and advice from cooking and swimming magazines. In the Islamic tradition, historically, there has been a strong emphasis on the person to person passing on of spiritual knowledge, and I think there is a virtue to this tradition. (Of course one should be cautious whom one seeks spiritual knowledge from and keep one's inner and outer eyes open, but it doesn't diminish the value of being able to take knowledge from another human.) It seems also that spiritual perception operates differently from person to person (for instance, for some people, it might be more direct whereas others more indirect, such as via intuition or dreams) and there isn't a one size fits all answer to be taken only from books, although books may have some things to offer.
At the same time, there are some basic things one can do regardless of what faith tradition one adheres to (if any) that aren't limited to Shi'ism, such as anything that develops one's sense of focus or reduction of unnecessary distractions in life, that are likely helpful along these lines. Also - as the narration says - most people have spiritual perception, especially people who already have an inclination towards religion and spirituality, and it doesn't hurt simply to pay attention to what is already there - especially in this busy day and age and age of secularism/materialism, a lot of people tend to just shut this sort of thing out.
Best wishes on your spiritual journey!