This type of grammatical construction is fairly common. However, your teacher was paraphrasing a bit.
会社に入る: To join a company
～になります: なる has many English translations. While 'to become' is the principal one provided, to get; to grow; to be; to reach; to attain are also valid translations depending on context.
Due to the use of the imperfect tense (generally used for present and future in English), 10年になります in the example sentence is indicating that the person is in their 10th year with the company, but not necessarily that 10 full years have passed.
The word for since (から) is omitted from your sentence example, but is implied. Following are a few example sentences (taken from here) with ～になります, some omitting particles and some not:
「彼が死んでから十年になります。」'It's been 10 years since he died.'
「彼が日本に来て３年になります。」'It's been 3 years since he came to Japan.'
「私は来月16歳になります。」'I'll be 16 (years old) next month.'
「私がここに来てから１０年になります。」'It'll have been 10 years since I came here.'
「今度の4月で、私たちはここに2年住んでいることになります。」'This April, we will have been living here for 2 years.'
「私たちは来年で英語を５年間学んだ事になります。」'As of next year, we will have learned English for 5 years.'
Finally, the more literal translation I would give for the sentence in your example is:
'It will be 10 years since I joined the company.'
'I've been with the company for 10 years.'