It seems that Japanese has far more loanwords than any other language I've heard spoken. I understand that English is far-reaching and a global language, but are there many known reasons that English has had a huge linguistic influence on Japan in particular?
A large part of the reason for so many loanwords in Japanese is that it has a way of picking them up from just about every language it interacts with—much like English, as was mentioned in a comment to the original question.
Truth of the matter is, depending on how broadly you want to define it, you could say that every word outside of 大和言葉【やまとことば】 is in fact a loanword, as 漢語【かんご】 vocabulary is largely derived from Chinese readings of kanji, if not directly-imported words from classical Chinese texts.
Outside of Chinese, the next major batch to come in arrived with the Portuguese missionaries prior to Tokugawa expelling the foreigners, and is where Japanese got words such as パン and 煙草【たばこ】.
During the Tokugawa era the flow of new foreign words into the language was reduced to a trickle, however with the opening of the country and the start of the Meiji Restoration a great deal of interest was taken in foreign culture. Many new words were borrowed or coined to accommodate new facets of everyday life that were starting to appear.
Probably the biggest boom in importing foreign words has been since the end of World War II, between the constant presence of US military personnel exposing people to new vocabulary, Japanese businesses trying to expand into English-speaking countries, and the overall trendiness of outside culture generally being on the rise (much like how many foreigners learn a bit of Japanese because they're into anime). Ultimately, as long as the cultural climate remains amicable towards increased cultural imports the proportion of foreign vocabulary integrated into the Japanese language will continue to rise as well.