This experience leads Whorf to put forth a theory, along with his mentor, a guy named Sapir, that the language and words we use actually determines at least part of how we perceive the world. Most famously he went on to study the language of the Hopi and declared that because they lack true tense markers they have a cyclical and distinctly non-Western concept of time. This work was later refuted by another linguist, Ekkehart Malotki, who clearly showed that yes the Hopi of course have ways of talking about events that happened in the past and the future. They also have a more cyclical, distinctly non-Western concept of time because most peoples besides industrialized Western peoples have a cyclical distinctly non-Western concept of time.
Despite this setback Whorf's work has continued to be influential in the study of linguistics and perception. There have been several studies about the effects one's language has on color perception. You can read about a few here and here. Basically the idea is that if you come from a culture with different color words you will actually, pre-attentively, see these colors as different from people who lack those color words. (Pre-attentively essentially means that it's not an artifact of you consciously deciding "oh, that's this color and not that color" but an actual perception difference at an unconscious level). For a long time I couldn't quite get my head around how this actually feels to the person perceiving the color but then I read something that pointed out that in English we have the word "pink." Now if we follow all our other color terms "pink" should just be "light red." It is just red with white added the way light green would be dark green with some white added. But instead we have the word "pink." That finally got me to understand this phenomenon. It does seem to me that I do see "pink" as a distinctive color from "red" in a very different way from how I perceive something like "green" and "light green."
When I started looking into Whorf's theories I found that they had turned up in recent media. A linguist named Keith Chen has put forth a theory that the way your language marks the future tense has an effect on your behavior. You can read about it here. Basically the theory is that if your language, like English, has a distinct split between present and future (i.e. we have two words for present and future and we have distinct tense markers) you are more likely to engage in behaviors that do not consider the future in favor the present such as not saving money, eating unhealthy foods, etc. His data seem to show that people who speak languages in which the future is marked as a continuation of the present, i.e. not entirely separate from the present, are better at saving and preparing for the future and healthier in general.
The article on Language Log presented above makes this point but when I read this I thought "Hrm, there seems to be more going on here." We'll probably never quite know for sure but the point I came to and the point the article makes is that these complex social behaviors (saving money or not saving money, eating healthy or not) are products of culture. It's pretty easy to find lots of correlations in a culture or across cultures. As an incredibly silly example for the purpose of illustration: a lot of people who are native Japanese speakers also live on an island but I don't think any linguists or anthropologists would want to make a causal link between living on an island and speaking Japanese. It is entirely likely that the above example is the same: there are certain cultures which may value the future in a different way and there going to be other differences that may look like they correlate but actually just correlate also with culture.
It seems like an interesting idea that the language we speak has some effect on how we perceive the world and our daily behaviors but when it comes down to it these are likely all artifacts of culture more generally and not necessarily language alone. Then again it's probably impossible to parse language apart from culture.