Translation:We are British.
The previous answers are mostly correct, but to answer quite precisely:
英 does mean "brave." It was apparently chosen as a transliteration for the country because "yīng" sounds like the first syllable of England, but literally 英国 still translates to "brave country".
美 means "beautiful," which is rather different from "nice." 美国 literally means "beautiful country." Some country names, like 加拿大 (Canada) and 意大利 (Italy), for example, are pure transliterations. But in the cases of England and America, the translations of "brave country" and "beautiful country" are also literally accurate - although you'll never see them translated this wsy.
英国人 by itself can mean either "English person" or "English people;" it's not clear from the characters by themselves. We only know it's plural in this context becuse the subject, 我们, is plural.
(I learned all of this from my first year of university-level Chinese, which I took way back in 1985, and have been studying ever since. Not from Google Translate.) :-)
It is more like transcription. The most typical translation of western country is italy, in chinese 意大利，yi4 da4 li4, if you read it, you will find it sounds like idaly->italy. America 美国, full name known as 美利坚合众国，mei3 li4 jian1 he2 zhong4 guo2, meili sounds like 'meri' in the word A'meri'ca. From history aspect, after the year 1860, self-strengthening movement began, famous politician called Zeng guofan, merge 米利坚(mi3li4jian1) with 美理哥(mei3li3ge1) and changed to 美利坚. I found ref in an article, it said by Zeng, "in all of the foreign countries of the Atlantic Ocean, England is the most cunning country, France ranks the second, Russian's force is greater than England and France,...美利坚人(American)，its nature is mellow, and always appears as respectful and submissive". Zeng wrote these in Chinese, I just roughly translated it. I think because of some history reason, we use more graceful character like 美(beautiful) to name America. Moreover, in japan, america is translated to 米国, in chinese literally means 'rice country', obviously, 米 mi3 is more related with the sound. For England 英国，when italian prieat matteo Ricci first introduce England to China, he used 谙戈利亚(an1ge3li4ya4), then England translated as 英圭黎(ying1gui1li2) , then 英吉利(ying1ji2li4), then England become united kingdom, 英吉利国 short for 英国.
My understanding is that 英国人 can mean either "English" or "British," and 英国 can refer to either England or Great Britain/the UK (although I think of it as England, but that's probably just me personally). Just like in English, Great Britain is made up of separate country names: 英格蘭 (yīnggélán - England), 蘇格蘭 (sūgélán - Scotland), 愛爾蘭 (aìěrlán - Ireland; add 北 at the beginning for Northern Ireland), and 威爾士 (weīěrshì - Wales). You can add 人 to the end of any of these country names to obtain the individual nationalities, although for an Englishman (or woman!), I think that 英国人 will be far more commonly used than 英格兰人.
The name 大不列顛 (dàbùlièdiān) "Great Britain" also exists, although I'm not sure how widely it's used these days outside of history books and atlases. Maybe a native Chinese speaker can tell us that.
I think 大不列颠 (Big Britain) is not used in daily conversation btw ppl, this kind of word usually used in history book. Nobody will call Britain 大不列颠 in daily life. History: Kingdom of Great Britain, 大不列颠王国（1707-1801），United kingdom of Great Britain and Ireland 大不列颠及爱尔兰联合王国(1801-1922), United Kingdom大不列颠及北爱尔兰联合王国(1922 till now)
'Chinese guys' was probably not the original primary translation, but accepted when a User suggested it as a translation in a previous exercise. It seems that no one, until now, has suggested it for this particular exercise. You should always report a new translation if you feel it is acceptable.